Semiparametric Discrete Choice Models for Bundles

abstractWe propose two approaches to estimate semiparametric discrete choice models for bundles. Our first approach is a kernel-weighted rank estimator based on a matching-based identification strategy. We establish its complete asymptotic properties and prove the validity of the nonparametric bootstrap for inference. We then introduce a new multi-index least absolute deviations (LAD) estimator as an alternative, of which the main advantage is its capacity to estimate preference parameters on both alternative- and agent-specific regressors. Both methods can account for arbitrary correlation in disturbances across choices, with the former also allowing for interpersonal heteroskedasticity. We also demonstrate that the identification strategy underlying these procedures can be extended naturally to panel data settings, producing an analogous localized maximum score estimator and a LAD estimator for estimating bundle choice models with fixed effects. We derive the limiting distribution of the former and verify the validity of the numerical bootstrap as an inference tool. All our proposed methods can be applied to general multi-index models. Monte Carlo experiments show that they perform well in finite samples. {{\em JEL classification\/}: C13, C14, C35.} {{\em Keywords\/}: Multi-index models; Rank correlation; Maximum score; Least absolute deviations; Panel data.}

notationAll vectors are column vectors. $\mathbb{R}^{p}$ is a $p$-dimensional Euclidean space equipped with the Euclidean norm $\Vert \cdot \Vert $, and $ \mathbb{R}_{+}\equiv \{x\in \mathbb{R}|x\geq 0\}$. We reserve letters $i$ and $m$ for indexing agents, $j$ and $l$ for indexing alternatives, and $s$ and $t$ for indexing time periods. The first element of a vector $v$ is denoted by $v^{(1)}$ and the sub-vector comprising its remaining elements is denoted by $\tilde{v}$. $P(\cdot )$ and $\mathbb{E}[\cdot ]$ denote probability and expectation, respectively. $1[\cdot ]$ is an indicator function that equals 1 when the event in the brackets occurs, and 0 otherwise. For two random vectors $U$ and $V$, the notation $U\overset{d}{=}V|\cdot $ means that $U$ and $V$ have identical distribution conditional on $\cdot $, and $U\perp V|\cdot $ means that $U$ and $V$ are independent conditional on $\cdot $. Symbols $\setminus$, $^{\prime }$, $\Leftrightarrow $, $\Rightarrow$, $\propto$, $\overset{d}{\rightarrow }$, and $\overset{p}{\rightarrow }$ represent set difference, matrix transposition, if and only if, implication, proportionality, convergence in distribution, and convergence in probability, respectively. For any (random) positive sequences $\{a_{N}\}$ and $\{b_{N}\}$, $a_{N}=O(b_{N})$ ($O_{p}(b_{N})$) means that $a_{N}/b_{N}$ is bounded (bounded in probability) and $a_{N}=o(b_{N})$ ($o_{p}(b_{N})$) means that $a_{N}/b_{N}\rightarrow0$ ($a_{N}/b_{N} \overset{p}{\rightarrow}0$).

itemize• (i) $\{(Y_{i},Z_{i})\}_{i=1}^{N}$ are i.i.d. across $i$, (ii) $(\epsilon _{1},\epsilon _{2},\eta )\perp Z$, and (iii) the joint distribution of $(\epsilon _{1},\epsilon _{2},\eta )$ is absolutely continuous on $\mathbb{R}^{2}\times \mathbb{R}_{+}$. • For any pair of $(i,m)$ and $j=1,2$, denote $ X_{imj}=X_{ij}-X_{mj}$, $W_{im}=W_{i}-W_{m}$, and $V_{im}(\beta)= V_{i}(\beta)-V_{m}(\beta)$. Then, (i) $X_{im1}^{(1)}$ ($ X_{im2}^{(1)}$) has almost everywhere (a.e.) positive Lebesgue density on $ \mathbb{R}$ conditional on $\tilde{X}_{im1}$ ($\tilde{X}_{im2}$) and conditional on $(X_{im2},W_{im})$ ($(X_{im1},W_{im})$) in a neighborhood of $ (X_{im2},W_{im})$ ($(X_{im1},W_{im})$) near zero, (ii) Elements in $X_{im1}$ ($ X_{im2}$), conditional on $(X_{im2},W_{im})$ ($(X_{im1},W_{im})$) in a neighborhood of $(X_{im2},W_{im})$ ($(X_{im1},W_{im})$) near zero, are linearly independent, (iii) $W_{im}^{(1)}$ has a.e. positive Lebesgue density on $\mathbb{R}$ conditional on $\tilde{W}_{im}$ and conditional on $V_{im}(\beta)$ in a neighborhood of $V_{im}(\beta)$ near zero, and (iv) Elements in $W_{im}$, conditional on $V_{im}(\beta)$ in a neighborhood of $V_{im}(\beta)$ near zero, are linearly independent. • $(\beta,\gamma)\in \mathcal{B}\times\mathcal{R}$, where $\mathcal{B}=\{b\in\mathbb{R}^{k_1}|b^{(1)}=1,\Vert b\Vert\leq C_b\}$ and $\mathcal{R}=\{r\in\mathbb{R}^{k_2}|r^{(1)}=1,\Vert r\Vert\leq C_r\}$ for some constants $C_b,C_r>0$. • $F_{1}(\cdot ,\cdot )$, $F_{2}(\cdot ,\cdot )$, and $ F_{b}(\cdot )$ are respectively $\mathbb{R}^{2}\mapsto \mathbb{R}$, $\mathbb{R}^{2}\mapsto \mathbb{R}$, and $\mathbb{R}\mapsto \mathbb{R}$ functions strictly increasing in each of their arguments. $F_{b}(0)=0$.

theoremSuppose Assumptions C1--C4 hold. Then $\beta $ and $\gamma $ are identified.

remarkThe matching-based method discussed here is distribution-free, accommodating arbitrary correlation among $(\epsilon_{1},\epsilon_{2},\eta)$ and interpersonal heteroskedasticity. Furthermore, as demonstrated in Sections (ref)--(ref), the resulting estimator is computationally tractable and has an asymptotic normal distribution, rendering inference relatively straightforward, particularly when employing bootstrap methods. However, this method has two main limitations. Firstly, it becomes difficult to apply when the model has many covariates, as this decreases the number of potential matches. Secondly, in cases where “common regressors” are present, it cannot estimate coefficients related to these variables unless their coefficients are assumed to be not “alternative-specific”.

itemize• $K(\cdot )$ is continuously differentiable and assumed to satisfy: (i) $\sup_{\upsilon }|K(\upsilon )|<\infty $, (ii) $\int K(\upsilon )\text{d}\upsilon =1$, (iii) for any positive integer $\iota \leq \max \{ \kappa _{\beta },\kappa _{\gamma } \}$ where $\kappa_\beta$ is an even integer greater than $k_1+k_2$ and $\kappa_\gamma$ is an even integer greater than 2, $\int \upsilon ^{\iota }K(\upsilon )\text{d}\upsilon =0$ if $\iota < \max\{\kappa_{\beta},\kappa_{\gamma}\}$, $\int \upsilon ^{\iota }K(\upsilon )\text{d}\upsilon\neq 0$ if $\iota=\max\{\kappa_{\beta},\kappa_{\gamma}\}$, and $\int \vert \upsilon ^{\max\{\kappa_{\beta},\kappa_{\gamma }\} }\vert K(\upsilon )\text{d}\upsilon <\infty $.\footnote{Here we use the same kernel function $K(\cdot)$ for estimating $\beta$ and $\gamma$ to avoid additional notations. However, in practice, one can use kernel functions with different orders by replacing $\max\{\kappa_{\beta},\kappa_{\gamma}\}$ with $\kappa_{\beta}$ and $\kappa_{\gamma}$,respectively. } • $h_{N}$ and $\sigma _{N}$ are sequences of positive numbers such that as $N\rightarrow \infty $: (i) $h_{N}\rightarrow 0$, $ \sigma _{N}\rightarrow 0$, (ii) $\sqrt{N}h_{N}^{k_{1}+k_{2}}\rightarrow \infty $, $\sqrt{N}\sigma _{N}^{2}\rightarrow \infty $, and (iii) $\sqrt{N} h_{N}^{\kappa _{\beta }}\rightarrow 0$, $\sqrt{N}\sigma _{N}^{\kappa _{\gamma }}\rightarrow 0$.

theoremIf Assumptions C1--C7 hold, then we have \begin{itemize} • $\sqrt{N}(\hat{\beta}-\beta )\overset{d}{\rightarrow } N(0,4\left\{ \mathbb{E}[\nabla ^{2}\varrho _{i}(\beta )]\right\} ^{-1} \mathbb{E}[\nabla \varrho _{i}(\beta )\nabla \varrho _{i}(\beta )^{\prime }]\left\{ \mathbb{E}[\nabla ^{2}\varrho _{i}(\beta )]\right\} ^{-1})$ or, alternatively, $\hat{\beta}-\beta $ has the linear representation: \begin{equation*} \hat{\beta}-\beta =-\frac{2}{N}\left\{ \mathbb{E}[\nabla ^{2}\varrho _{i}(\beta )]\right\} ^{-1}\sum_{i=1}^{N}\varrho _{i}(\beta )+o_{p}(N^{-1/2}), \end{equation*} with $\varrho _{i}\left( \cdot \right) $ defined in ((ref)). • $\sqrt{N}(\hat{\gamma}-\gamma )\overset{d}{\rightarrow }N\left( 0,\mathbb{E}[\nabla ^{2}\tau _{i}(\gamma )]^{-1}\mathbb{E}\left[ \varDelta _{i}\varDelta_{i}^{\prime }\right] \mathbb{E}[\nabla ^{2}\tau _{i}(\gamma )]^{-1})\right) $, where \begin{equation*} \varDelta_{i}\equiv 2\nabla \tau _{i}(\gamma )+2\mathbb{E}\left[ \nabla _{13}^{2}\mu \left( V_{i}(\beta ),V_{i}(\beta ),\gamma \right) \right] \left\{ \mathbb{E}[\nabla ^{2}\varrho _{i}(\beta )]\right\} ^{-1}\varrho _{i}(\beta ), \end{equation*} with $\tau _{i}\left( \cdot \right) $ and $\mu \left( \cdot ,\cdot ,\cdot \right) $ defined in ((ref)) and ((ref)), respectively, and $\nabla _{13}^{2}$ denotes the second order derivative w.r.t. the first and third arguments. \end{itemize}

comment\textcolor{red}{ Although the expressions for $V_{\beta}$, $V_{\gamma}$, $\Psi$, and $\Lambda$ are quite involved, note that $V_{\beta}$ and $V_{\gamma}$ represent the second derivatives of the limit of the expectation of the maximands in ((ref)) and ((ref)), respectively; and $\Psi$ and $\Lambda$ represent the variances of the limit of their projections. The latter takes into account the fact that ((ref)) uses estimated $\beta$. In practice, statistical inference, such as confidence regions, can be conducted using the standard nonparametric bootstrap. }

theoremSuppose Assumptions C1--C7 hold. Then, we have \[ \sqrt{N} ( \hat{\beta}^{\ast}-\hat{\beta} ) \overset{d}{\rightarrow }N ( 0,4\left\{ \mathbb{E}[\nabla^{2}\varrho_{i}(\beta)]\right\} ^{-1}\mathbb{E}[\nabla\varrho_{i}(\beta)\nabla\varrho_{i}(\beta)^{\prime }]\left\{ \mathbb{E}[\nabla^{2}\varrho_{i}(\beta)]\right\} ^{-1} ) \textrm{ and} \] \[ \sqrt{N}\left( \hat{\gamma}^{\ast}-\hat{\gamma}\right) \overset{d}{\rightarrow}N\left( 0,\mathbb{E}[\nabla^{2}\tau_{i}(\gamma )]^{-1}\mathbb{E}\left[ \varDelta_{i}\varDelta_{i}^{\prime}\right] \mathbb{E}[\nabla^{2}\tau_{i}(\gamma)]^{-1}\right) \] conditional on the sample.

theoremSuppose Assumptions C1-C7 hold. Then, under $\mathbb{H} _{0},$ \begin{equation*} \sqrt{N} ( \mathcal{\hat{L}}_{N}^{K}\left( \hat{\gamma}\right) -\mathcal{ \bar{L}}\left( \gamma \right) ) \overset{d}{\rightarrow }N\left( 0,\Delta \right) . \end{equation*}

itemize• A wrong prediction incurs a loss of 2, i.e., $q_{(1,0)}(Z_{i} ,Z_{m},\vartheta)=2$ when $\{I^{+}_{im(1,0)}(\vartheta)=1,\Delta p_{(1,0)}(Z_{i},Z_{m})\leq0\}$ or $\{I^{-}_{im(1,0)}(\vartheta)=1,\Delta p_{(1,0)}(Z_{i},Z_{m})\geq0\}$ occurs. • A correct prediction or no prediction incurs 0 loss, i.e., $q_{(1,0)}(Z_{i},Z_{m},\vartheta)=0$ when $\{I^{+}_{im(1,0)}(\vartheta )=1,\Delta p_{(1,0)}(Z_{i},Z_{m})\geq0\}$ or $\{I^{-}_{im(1,0)}(\vartheta )=1,\Delta p_{(1,0)}(Z_{i},Z_{m})\leq0\}$ or $\{I^{+}_{im(1,0)}(\vartheta )=I^{-}_{im(1,0)}(\vartheta)=0\}$ occurs.

itemize• (i) $\{(Y_{i},Z_{i})\}_{i=1}^{N}$ are i.i.d. across $i$, (ii) $(\epsilon_{1},\epsilon_{2},\eta)\perp Z$, and (iii) the joint distribution of $(\epsilon_{1},\epsilon_{2},\eta)$ is absolutely continuous on $\mathbb{R}^{2}\times\mathbb{R}_{+}$. • For any pair of $(i,m)$ and $j=1,2$, denote $X_{imj} =X_{ij}-X_{mj}$, $W_{im}=W_{i}-W_{m}$, and $S_{im}= S_{i}-S_{m}$. Then, (i) $X_{im1}^{(1)}$ ($X_{im2}^{(1)}$) has a.e. positive Lebesgue density on $\mathbb{R}$ conditional on $(\tilde{X}_{im1},X_{im2},W_{im},S_{im})$ ($(\tilde{X}_{im2},X_{im1},W_{im},S_{im})$), (ii) Elements in $(X_{im1},S_{im})$ ($(X_{im2},S_{im})$), conditional on $(X_{im2},W_{im})$ ($(X_{im1} ,W_{im})$), are linearly independent, (iii) $W_{im}^{(1)}$ has a.e. positive Lebesgue density on $\mathbb{R}$ conditional on $(\tilde{W} _{im},X_{im1},X_{im2},S_{im})$, and (iv) Elements in $(W_{im},S_{im})$, conditional on $(X_{im1},X_{im2})$, are linearly independent. • $\theta=(\beta,\gamma,\rho_{1},\rho_{2},\rho_{b})\in \Theta\subset\mathbb{R}^{k_{1}+k_{2}+3k_{3}}$, where $\Theta=\{\vartheta =(b,r,\varrho_{1},\varrho_{2},\varrho_{b})|b^{(1)}=r^{(1)}=1,\Vert \vartheta\Vert\leq C\}$ for some constant $C>0$. • $F_{1}(\cdot,\cdot)$, $F_{2}(\cdot,\cdot)$, and $F_{b}(\cdot)$ are respectively $\mathbb{R}^{2}\mapsto\mathbb{R}$, $\mathbb{R}^{2}\mapsto\mathbb{R}$, and $\mathbb{R}\mapsto\mathbb{R}$ functions strictly increasing in each of their arguments. $F_{b}(0)=0$.

theoremUnder Assumptions CL1--CL4, $Q_{(1,0)}(\theta)<Q_{(1,0)}(\vartheta)$ for all $\vartheta\in\Theta \setminus\{\theta\}$.

itemize• $\sup_{(z_{i},z_{m})\in\mathcal{Z}^{2}}\vert\Delta\hat{p}_{(1,0)}(z_{i},z_{m})-\Delta p_{(1,0)}(z_{i},z_{m})\vert\overset{p}{\rightarrow}0$ as $n\rightarrow\infty$, where $\mathcal{Z}$ is the support of $Z$.

theoremSuppose Assumptions CL1--CL5 hold. We have $\hat{\theta}\overset{p}{\rightarrow}\theta$.

remarkIn addition to ((ref)) and ((ref)), we can also construct population and sample criterion functions based on the following inequalities: \begin{align} \{X_{im1}^{\prime}\beta+S^{\prime}_{im}\rho_{1}\leq0,X_{im2}^{\prime} \beta+S_{im}^{\prime}\rho_{2}\geq0,W_{im}^{\prime}\gamma+S^{\prime}_{im}\rho _{b}\leq0\} & \Rightarrow\Delta p_{(0,1)}(Z_{i},Z_{m})\geq 0,\\ \{X_{im1}^{\prime}\beta+S^{\prime}_{im}\rho_{1}\geq0,X_{im2}^{\prime} \beta+S_{im}^{\prime}\rho_{2}\leq0,W_{im}^{\prime}\gamma+S^{\prime}_{im}\rho _{b}\geq0\} & \Rightarrow\Delta p_{(0,1)}(Z_{i},Z_{m})\leq 0,\\ \{X_{im1}^{\prime}\beta+S^{\prime}_{im}\rho_{1}\geq0,X_{im2}^{\prime} \beta+S_{im}^{\prime}\rho_{2}\geq0,W_{im}^{\prime}\gamma+S^{\prime}_{im}\rho _{b}\geq0\} & \Rightarrow\Delta p_{(1,1)}(Z_{i},Z_{m})\geq 0,\\ \{X_{im1}^{\prime}\beta+S^{\prime}_{im}\rho_{1}\leq0,X_{im2}^{\prime} \beta+S_{im}^{\prime}\rho_{2}\leq0,W_{im}^{\prime}\gamma+S^{\prime}_{im}\rho _{b}\leq0\} & \Rightarrow\Delta p_{(1,1)}(Z_{i},Z_{m})\leq0, \\ \{X_{im1}^{\prime}\beta+S^{\prime}_{im}\rho_{1}\leq0,X_{im2}^{\prime} \beta+S_{im}^{\prime}\rho_{2}\leq0,W_{im}^{\prime}\gamma+S^{\prime}_{im}\rho _{b}\leq0\} & \Rightarrow\Delta p_{(0,0)}(Z_{i},Z_{m})\geq 0,\\ \{X_{im1}^{\prime}\beta+S^{\prime}_{im}\rho_{1}\geq0,X_{im2}^{\prime} \beta+S_{im}^{\prime}\rho_{2}\geq0,W_{im}^{\prime}\gamma+S^{\prime}_{im}\rho _{b}\geq0\} & \Rightarrow\Delta p_{(0,0)}(Z_{i},Z_{m})\leq 0, \end{align} where $\Delta p_{d}(Z_{i},Z_{m})$ for $d\in\{(0,1),(1,1),(0,0)\}$ are similarly defined as $\Delta p_{(1,0)}(Z_{i},Z_{m})$ in ((ref))--((ref)). Note that the following two events are not informative in the sense that they are not sufficient to predict the sign of any $\Delta p_{d}(Z_{i},Z_{m})$: \begin{align*} & \{X_{im1}^{\prime}\beta+S^{\prime}_{im}\rho_{1}\le0,X_{im2}^{\prime} \beta+S_{im}^{\prime}\rho_{2}\leq0,W_{im}^{\prime}\gamma+S^{\prime}_{im}\rho _{b}\geq0\},\\ & \{X_{im1}^{\prime}\beta+S^{\prime}_{im}\rho_{1}\geq0,X_{im2}^{\prime} \beta+S_{im}^{\prime}\rho_{2}\geq0,W_{im}^{\prime}\gamma+S^{\prime}_{im}\rho _{b}\leq0\}. \end{align*} For any $d\in\{(0,1),(1,1),(0,0)\}$ corresponding to ((ref) )--((ref)), ((ref))--((ref)), and ((ref))--((ref)), respectively, $(I_{imd} ^{+}(\vartheta),I_{imd}^{-}(\vartheta))$ can be defined analogously to $(I_{im(1,0)}^{+}(\vartheta),I_{im(1,0)}^{-}(\vartheta))$ in ((ref))--((ref)). Then, any single criterion $Q_{d}(\vartheta)=\mathbb{E}[q_{d}(Z_{i},Z_{m},\vartheta)]$ ($Q_{d} ^{D}(\vartheta)=\mathbb{E}[q_{d}^{D}(Z_{i},Z_{m},\vartheta)]$) or their convex combination can be used to identify $\theta$. For estimation purposes, we recommend to use a sample criterion constructed by combining the sample analogues of all available population criteria (e.g., summing them up), to achieve higher finite sample efficiency. For example, in the simulation studies presented in Section (ref), we compute \[ \hat{\theta}=\arg\min_{\vartheta\in\Theta}\sum_{i=1}^{N-1}\sum_{m>i}\sum _{d\in\mathcal{D}}\hat{q}_{imd}^{D}(\vartheta), \] where $\mathcal{D}=\{(0,0),(1,0),(0,1),(1,1)\}$ and $\hat{q}_{imd}^{D}(\vartheta)$ is similarly defined as in ((ref)).

enumerate• (i) $\{(Y_{i}^{T},Z_{i}^{T})\}_{i=1}^{N}$ are i.i.d, where $Y_{i}^{T}\equiv \{(Y_{i(0,0)t},Y_{i(1,0)t},Y_{i(0,1)t},Y_{i(1,1)t})\}_{t=1}^T$, (ii) for almost all $ (\alpha ,Z^{T})$, $\xi _{t}\overset{d}{=}\xi _{s}|\left( \alpha ,Z_{t},Z_{s}\right) $, and (iii) the distribution of $\xi _{t}$ conditional on $\alpha $ is absolutely continuous w.r.t. the Lebesgue measure on $ \mathbb{R}^{2}\times \mathbb{R}_{+}$ for all $t$. • $(\beta,\gamma)\in \mathcal{B}\times \mathcal{R}$, where $\mathcal{B}=\{b\in \mathbb{R} ^{k_{1}}|\left\Vert b\right\Vert =1,b^{(1)}\neq 0\}$ and $\mathcal{R}=\{r\in \mathbb{R}^{k_{2}}|\left\Vert r\right\Vert =1,r^{(1)}\neq 0\}$.\footnote{We adopt the same scale normalization as in KimPollard1990 and SeoOtsu2018. This simplifies the asymptotic analysis of our MS estimators, enabling us to use the results established in these works directly. It is easy to verify that the normalization used here is equivalent to that in Assumption C3.} • (i) $X_{1ts}^{(1)}$ ($X_{2ts}^{(1)}$) has a.e. positive Lebesgue density on $\mathbb{R}$ conditional on $\tilde{X}_{1ts}$ ($\tilde{X} _{2ts}$) and conditional on $\left( X_{2ts},W_{ts}\right)$ ($\left( X_{1ts},W_{ts}\right) $) in a neighborhood of $\left( X_{2ts},W_{ts}\right)$ ($\left( X_{1ts},W_{ts}\right)$) near zero, (ii) the support of $X_{1ts}$ ($X_{2ts}$), conditional on $\left( X_{2ts},W_{ts}\right)$ ($\left( X_{1ts},W_{ts}\right)$) in a neighborhood of $\left( X_{2ts},W_{ts}\right)$ ($\left( X_{1ts},W_{ts}\right)$) near zero, is not contained in any proper linear subspace of $\mathbb{R}^{k_{1}}$, (iii) $W_{ts}^{(1)}$ has a.e. positive Lebesgue density on $\mathbb{R}$ conditional on $\tilde{W}_{ts}$ and conditional on $\left( X_{1ts},X_{2ts}\right) $ in a neighborhood of $\left( X_{1ts},X_{2ts}\right) $ near zero, and (iv) the support of $W_{ts}$, conditional on $\left( X_{1ts},X_{2ts}\right) $ in a neighborhood of $\left( X_{1ts},X_{2ts}\right) $ near zero, is not contained in any proper linear subspace of $\mathbb{R} ^{k_{2}}$. • $F_{j}\left( \cdot ,\cdot ,\cdot \right) ,$ $j=1,2,$ and $ F_{b}\left( \cdot \right) $ are strictly increasing in their arguments. $F_{b}(0)=0$.

theoremSuppose Assumptions P1--P4 hold. Then $\beta$ and $ \gamma $ are identified.

enumerate• Let $f_{X_{jt},W_{t}}(\cdot)$ denote the density of the random vector $(X_{jt},W_{t})$ for all $j=1,2$, and $f_{X_{1ts},X_{2ts}}(\cdot)$ denote the density of the random vector $ \left( X_{1ts},X_{2ts}\right)$. $f_{X_{jt},W_{t}}(\cdot)$'s and $ f_{X_{1ts},X_{2ts}}(\cdot)$ are absolutely continuous, bounded from above on their supports, strictly positive in a neighborhood of zero, and twice continuous differentiable a.e. with bounded derivatives. • For all $d\in\mathcal{D}$, $\mathbb{E}[Y_{dts}\text{sgn} \left( X_{1ts}^{\prime}b\right) |X_{2ts},W_{ts}]$ and $\mathbb{E}[Y_{dts} \text{sgn}\left( X_{2ts}^{\prime}b\right) |X_{1ts},W_{ts}]$ are twice continuously differentiable w.r.t. $b$ a.e. with bounded derivatives, and $ \mathbb{E}[Y_{dts}\text{sgn}\left( W_{ts}^{\prime}r\right) |X_{1ts},X_{2ts}]$ is twice continuously differentiable w.r.t. $r$ a.e. with bounded derivatives. • $K(\cdot)$ is a function of bounded variation and has a compact support. It satisfies: (i) $K(v)\geq0$ and $\sup_{v}|K(v)|<\infty$, (ii) $\int K(v)$d$v=1$, (iii) $\int vK(v)$d$v=0$, (iv) $\int v^{2}K(v)$d$ v<\infty$ and (v) $K(\cdot)$ is twice continuously differentiable with bounded first and second derivatives. • $(\hat{\beta},\hat{\gamma})$ satisfies \begin{equation*} N^{-1}\mathcal{L}_{N,\beta}^{P,K}(\hat{\beta})\geq\max_{b\in\mathcal{B} }N^{-1}\mathcal{L}_{N,\beta}^{P,K}(b)-o_{p}((Nh_{N}^{k_{1}+k_{2}})^{-2/3}) \end{equation*} and \begin{equation*} N^{-1}\mathcal{L}_{N,\gamma}^{P,K}(\hat{\gamma})\geq\max_{r\in\mathcal{R} }N^{-1}\mathcal{L}_{N,\gamma}^{P,K}(r)-o_{p}((N\sigma_{N}^{2k_{1}})^{-2/3}). \end{equation*} • $h_{N}$ and $\sigma_{N}$ are sequences of positive numbers such that as $N\rightarrow\infty$: (i) $h_{N}\rightarrow0$ and $ \sigma _{N}\rightarrow0$, (ii) $Nh_{N}^{k_{1}+k_{2}}\rightarrow\infty$ and $ N\sigma_{N}^{2k_{1}}\rightarrow\infty,$ and (iii) $ (Nh_{N}^{k_{1}+k_{2}})^{2/3}h_{N}^{2}\rightarrow0$ and $(N \sigma_{N}^{2k_{1}})^{2/3}\sigma_{N}^{2}\rightarrow0$.

theoremSuppose Assumptions P1--P9 hold. Then, \begin{equation*} (Nh_{N}^{k_{1}+k_{2}})^{1/3}(\hat{\beta}-\beta)\overset{d}{\rightarrow} \arg\max_{\rho\in\mathbb{R}^{k_{1}}}\mathcal{Z}_{1}\left( \rho\right) , \end{equation*} where $\mathcal{Z}_{1}$ is a Gaussian process taking values in $\ell^{\infty }\left( \mathbb{R}^{k_{1}}\right) ,$ with $\mathbb{E}\left( \mathcal{Z} _{1}\left( \rho\right) \right) =\frac{1}{2}\rho^{\prime}\mathbb{V}\rho,$ covariance kernel $\mathbb{H}_{1}\left( \rho_{1},\rho_{2}\right) ,$ and $ \rho,\rho_{1},\rho_{2}\in\mathbb{R}^{k_{1}},$ and \begin{equation*} (N\sigma_{N}^{2k_{1}})^{1/3}\left( \hat{\gamma}-\gamma\right) \overset{d}{ \rightarrow}\arg\max_{\delta\in\mathbb{R}^{k_{2}}}\mathcal{Z}_{2}\left( \delta\right) , \end{equation*} where $\mathcal{Z}_{2}$ is a Gaussian process taking values in $\ell^{\infty }\left( \mathbb{R}^{k_{2}}\right) ,$ with $\mathbb{E}\left( \mathcal{Z} _{2}\left( \delta\right) \right) =\frac{1}{2}\delta^{\prime}\mathbb{W}\delta$ , covariance kernel $\mathbb{H}_{2}\left( \delta_{1},\delta _{2}\right) ,$ and $\delta,\delta_{1},\delta_{2}\in\mathbb{R}^{k_{2}}.$ $\mathbb{V}$, $ \mathbb{W}$, $\mathbb{H}_{1}$, and $\mathbb{H}_{2}$ are defined, respectively, by equations ((ref)), ((ref)), ((ref)), and ( (ref)) in Appendix (ref).

enumerate• $\varepsilon _{N1}$ and $\varepsilon _{N2}$ are sequences of positive numbers such that as $N\rightarrow \infty $: (i) $ \varepsilon _{N1}\rightarrow 0$ and $\varepsilon _{N2}\rightarrow 0$, (ii) $ N\varepsilon _{N1}\rightarrow \infty $ and $N\varepsilon _{N2}\rightarrow \infty $, (iii) $\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}}\rightarrow \infty $ and $\varepsilon _{N2}^{-1}\sigma _{N}^{2k_{1}}\rightarrow \infty $, and (iv) $(\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}})^{2/3}h_{N}^{2}\rightarrow 0 $ and $(\varepsilon _{N2}^{-1}\sigma _{N}^{2k_{1}})^{2/3}\sigma _{N}^{2}\rightarrow 0$.

theoremSuppose Assumptions P1--P10 hold. Then \begin{equation*} (\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}})^{1/3}(\hat{\beta}^{\ast }-\hat{ \beta})\overset{d}{\rightarrow }\arg \max_{\rho \in \mathbb{R}^{k_{1}}} \mathcal{Z}_{1}^{\ast }\left( \rho \right) \ conditional on the sample, \end{equation*} and \begin{equation*} (\varepsilon _{N2}^{-1}\sigma _{N}^{2k_{1}})^{1/3}\left( \hat{\gamma}^{\ast }-\hat{\gamma}\right) \overset{d}{\rightarrow }\arg \max_{\delta \in \mathbb{ R}^{k_{2}}}\mathcal{Z}_{2}^{\ast }\left( \delta \right) conditional on the sample, \end{equation*} where $\mathcal{Z}_{1}^{\ast }\left( \rho \right) $ and $\mathcal{Z} _{2}^{\ast }\left( \delta \right) $ are independent copies of $\mathcal{Z} _{1}\left( \rho \right) $ and $\mathcal{Z}_{2}\left( \delta \right) $ defined in Theorem (ref), respectively.

theoremSuppose Assumptions P1--P9 hold. Then, under $\mathbb{H} _{0},$ \begin{equation*} \sqrt{N\sigma _{N}^{2k_{1}}}( N^{-1}\mathcal{L}_{N,\gamma }^{P,K}(\hat{ \gamma})-\mathcal{\bar{L}}^{P}\left( \gamma \right) ) \overset{d}{ \rightarrow }N( 0,\Delta ^{P}_{\gamma}) , \end{equation*} where \begin{equation*} \Delta ^{P}_{\gamma}=\lim_{N\rightarrow \infty }Var\left( \sigma _{N}^{k_{1}}\sum_{t>s}\mathcal{K}_{\sigma _{N}}(X_{i1ts},X_{i2ts})(Y_{i(1,1)t}-Y_{i(1,1)s})sgn (W_{its}^{\prime }\gamma )\right) . \end{equation*}

itemize• We are comparing the same agent in different time periods, rather than different agents. • $\Delta p_{(1,0)}(Z_{t}, Z_{s})$ is defined as the difference between choice probabilities of $(1,0)$ in time periods $t$ and $s$, conditioning on all observable covariates from these two periods. This is because we do not impose any restriction on the way that $\alpha$ is dependent on $(Z_{t}, Z_{s})$, and thus we must keep both $Z_{t}$ and $Z_{s}$ in the conditioning sets when integrating out the $\alpha$ in ((ref)) and ((ref)).\footnote{Recall that in cross-sectional model, $\Delta p_{(1,0)}(Z_{i},Z_{m})\equiv P(Y_{(1,0)i}=1|Z_{i})-P(Y_{(1,0)m}=1|Z_{m})$ since $(Y_{i},Z_{i})$ and $(Y_{m},Z_{m})$ are assumed to be independent.}

enumerate• (i) $\{(Y_{i}^{T},Z_{i}^{T})\}_{i=1}^{N}$ are i.i.d, where $Y_{i}^{T}\equiv \{(Y_{i(0,0)t},Y_{i(1,0)t},Y_{i(0,1)t},Y_{i(1,1)t})\}_{t=1}^T$, (ii) for almost all $ (\alpha ,Z^{T})$, $\xi _{t}\overset{d}{=}\xi _{s}|\left( \alpha ,Z_{t},Z_{s}\right) $, and (iii) the distribution of $\xi _{t}$ conditional on $\alpha $ is absolutely continuous w.r.t. the Lebesgue measure on $ \mathbb{R}^{2}\times \mathbb{R}_{+}$ for all $t$. • $\theta=(\beta,\gamma,\rho_{1},\rho_{2},\rho_{b})\in \Theta\subset\mathbb{R}^{k_{1}+k_{2}+3k_{3}}$, where $\Theta=\{\vartheta =(b,r,\varrho_{1},\varrho_{2},\varrho_{b})|b^{(1)}=r^{(1)}=1,\Vert \vartheta\Vert\leq C\}$ for some constant $C>0$. • (i) $X_{1ts}^{(1)}$ ($X_{2ts}^{(1)}$) has a.e. positive Lebesgue density on $\mathbb{R}$ conditional on $(\tilde{X}_{1ts},X_{2ts},W_{ts},S_{ts})$ ($(\tilde{X}_{2ts},X_{1ts},W_{ts},S_{ts})$), (ii) the support of $(X_{1ts},S_{ts})$ ($(X_{2ts},S_{ts})$) is not contained in any proper linear subspace of $\mathbb{R}^{k_{1}+k_3}$ conditional on $(X_{2ts},W_{ts})$ ($(X_{1ts},W_{ts})$), (iii) $W_{ts}^{(1)}$ has a.e. positive Lebesgue density on $\mathbb{R}$ conditional on $(\tilde{W}_{ts},X_{1ts},X_{2ts},S_{ts})$, and (iv) the support of $(W_{ts},S_{ts})$ is not contained in any proper linear subspace of $\mathbb{R}^{k_{2}+k_3}$ conditional on $(X_{1ts},X_{2ts})$. • $F_{j}\left( \cdot ,\cdot ,\cdot \right) ,$ $j=1,2,$ and $ F_{b}\left( \cdot\right) $ are strictly increasing in their arguments. $F_{b}(0)=0$.

theoremUnder Assumptions PL1--PL4, $Q_{ts(1,0)}^P(\theta)<Q_{ts(1,0)}^P(\vartheta)$ for all $\vartheta\in\Theta \setminus\{\theta\}$.

itemize• $\sup_{(z_{t},z_{s})\in\mathcal{Z}_{t}\times\mathcal{Z}_{s}}\vert\Delta\hat{p}_{(1,0)}(z_{t},z_{s})-\Delta p_{(1,0)}(z_{t},z_{s})\vert\overset{p}{\rightarrow}0$ as $n\rightarrow\infty$ for all $t\neq s$, where $\mathcal{Z}_{t}$ is the support of $Z_{t}$.

theoremSuppose Assumptions PL1--PL5 hold. We have $\hat{\theta}\overset{p}{\rightarrow}\theta$.

appendix\section{Tables} \begin{table}[H] \caption{Design 1, Performance of MRC Estimators} \begin{tabular}{lccccccccc} \toprule & \multicolumn{4}{c}{$\beta$} & & \multicolumn{4}{c}{$\gamma$} \\ \cmidrule{2-5}\cmidrule{7-10} & MBIAS & RMSE & MED & MAD & & MBIAS & RMSE & MED & MAD \\ \cmidrule{2-5}\cmidrule{7-10} $N=250$ & 0.068 & 0.534 & 0.075 & 0.424 & & 0.101 & 0.452 & 0.074 & 0.318 \\ $N=500$ & 0.024 & 0.367 & 0.028 & 0.274 & & 0.065 & 0.314 & 0.038 & 0.208 \\ $N=1000$ & 0.024 & 0.237 & 0.016 & 0.154 & & 0.040 & 0.215 & 0.034 & 0.131 \\ \bottomrule \end{tabular} \end{table} \begin{table}[H] \caption{Design 1, Performance of Nonparametric Bootstrap} \begin{tabular}{lccccc} \toprule & \multicolumn{2}{c}{$\beta$} & & \multicolumn{2}{c}{$\gamma$} \\ \cmidrule{2-3}\cmidrule{5-6} & COVERAGE & LENGTH & & COVERAGE & LENGTH \\ \cmidrule{2-3}\cmidrule{5-6} $N=250$ & 0.835 & 2.505 & & 0.915 & 2.145 \\ $N=500$ & 0.862 & 1.719 & & 0.918 & 1.492 \\ $N=1000$ & 0.926 & 1.164 & & 0.938 & 1.002 \\ \bottomrule \end{tabular} \end{table} \begin{table}[H] \caption{Design 1, Performance of LAD Estimators} \begin{tabular}{lccccccccc} \toprule & \multicolumn{4}{c}{$\beta$} & & \multicolumn{4}{c}{$\gamma$} \\ \cmidrule{2-5}\cmidrule{7-10} & MBIAS & RMSE & MED & MAD & & MBIAS & RMSE & MED & MAD \\ \cmidrule{2-5}\cmidrule{7-10} $N=250$ & 0.102 & 0.274 & 0.090 & 0.180 & & 0.040 & 0.418 & 0.017 & 0.247 \\ $N=500$ & 0.112 & 0.213 & 0.114 & 0.144 & & 0.032 & 0.274 & 0.004 & 0.173 \\ $N=1000$ & 0.106 & 0.164 & 0.100 & 0.110 & & 0.030 & 0.189 & 0.022 & 0.124 \\ \midrule & & & & & & & & & \\ \midrule & \multicolumn{4}{c}{$\rho_1$} & & \multicolumn{4}{c}{$\rho_2$} \\ \cmidrule{2-5}\cmidrule{7-10} & MBIAS & RMSE & MED & MAD & & MBIAS & RMSE & MED & MAD \\ \cmidrule{2-5}\cmidrule{7-10} $N=250$ & 0.022 & 0.208 & 0.011 & 0.133 & & 0.019 & 0.200 & 0.011 & 0.128 \\ $N=500$ & 0.010 & 0.134 & 0.005 & 0.092 & & 0.009 & 0.135 & 0.001 & 0.089 \\ $N=1000$ & 0.010 & 0.095 & 0.007 & 0.062 & & 0.013 & 0.092 & 0.011 & 0.061 \\ \bottomrule \end{tabular} \end{table} \begin{table}[H] \caption{Design 2, Performance of MRC Estimators} \begin{tabular}{lccccccccc} \toprule & \multicolumn{4}{c}{$\beta$} & & \multicolumn{4}{c}{$\gamma$} \\ \cmidrule{2-5}\cmidrule{7-10} & MBIAS & RMSE & MED & MAD & & MBIAS & RMSE & MED & MAD \\ \cmidrule{2-5}\cmidrule{7-10} $N=250$ & 0.067 & 0.585 & 0.051 & 0.452 & & 0.083 & 0.471 & 0.040 & 0.325 \\ $N=500$ & 0.061 & 0.433 & 0.038 & 0.336 & & 0.072 & 0.328 & 0.058 & 0.216 \\ $N=1000$ & 0.029 & 0.283 & 0.015 & 0.182 & & 0.040 & 0.219 & 0.024 & 0.135 \\ \bottomrule \end{tabular} \end{table} \begin{table}[H] \caption{Design 2, Performance of Nonparametric Bootstrap} \begin{tabular}{lccccc} \toprule & \multicolumn{2}{c}{$\beta$} & & \multicolumn{2}{c}{$\gamma$} \\ \cmidrule{2-3}\cmidrule{5-6} & COVERAGE & LENGTH & & COVERAGE & LENGTH \\ \cmidrule{2-3}\cmidrule{5-6} $N=250$ & 0.870 & 2.847 & & 0.911 & 2.292 \\ $N=500$ & 0.894 & 1.992 & & 0.927 & 1.566 \\ $N=1000$ & 0.925 & 1.358 & & 0.933 & 1.067 \\ \bottomrule \end{tabular} \end{table} \begin{table}[H] \caption{Design 2, Performance of LAD Estimators} \begin{tabular}{lccccccccc} \toprule & \multicolumn{4}{c}{$\beta$} & & \multicolumn{4}{c}{$\gamma$} \\ \cmidrule{2-5}\cmidrule{7-10} & MBIAS & RMSE & MED & MAD & & MBIAS & RMSE & MED & MAD \\ \cmidrule{2-5}\cmidrule{7-10} $N=250$ & 0.109 & 0.312 & 0.102 & 0.205 & & 0.059 & 0.491 & 0.018 & 0.297 \\ $N=500$ & 0.107 & 0.222 & 0.103 & 0.147 & & 0.048 & 0.308 & 0.023 & 0.189 \\ $N=1000$ & 0.112 & 0.172 & 0.113 & 0.124 & & 0.028 & 0.209 & 0.013 & 0.143 \\ \midrule & & & & & & & & & \\ \midrule & \multicolumn{4}{c}{$\rho_1$} & & \multicolumn{4}{c}{$\rho_2$} \\ \cmidrule{2-5}\cmidrule{7-10} & MBIAS & RMSE & MED & MAD & & MBIAS & RMSE & MED & MAD \\ \cmidrule{2-5}\cmidrule{7-10} $N=250$ & -0.042 & 0.209 & -0.050 & 0.139 & & -0.036 & 0.211 & -0.047 & 0.154 \\ $N=500$ & -0.034 & 0.146 & -0.040 & 0.098 & & -0.037 & 0.147 & -0.037 & 0.098 \\ $N=1000$ & -0.041 & 0.105 & -0.043 & 0.074 & & -0.037 & 0.100 & -0.039 & 0.073 \\ \bottomrule \end{tabular} \end{table} \begin{table}[H] \caption{Design 3, Performance of MS Estimators} \begin{tabular}{lccccccccc} \toprule & \multicolumn{4}{c}{$\beta$} & & \multicolumn{4}{c}{$\gamma$} \\ \cmidrule{2-5}\cmidrule{7-10} & MBIAS & RMSE & MED & MAD & & MBIAS & RMSE & MED & MAD \\ \cmidrule{2-5}\cmidrule{7-10} $N=1000$ & -0.008 & 0.371 & 0.003 & 0.244 & & -0.014 & 0.411 & -0.057 & 0.276 \\ $N=2500$ & 0.006 & 0.332 & 0.003 & 0.236 & & -0.004 & 0.375 & -0.040 & 0.246 \\ $N=5000$ & -0.002 & 0.291 & 0.001 & 0.235 & & 0.009 & 0.367 & -0.016 & 0.252 \\ \bottomrule \end{tabular} \end{table} \begin{table}[H] \caption{Design 3, Performance of Numerical Bootstrap} \begin{tabular}{lccccc} \toprule & \multicolumn{2}{c}{$\beta$} & & \multicolumn{2}{c}{$\gamma$} \\ \cmidrule{2-3}\cmidrule{5-6} & COVERAGE & LENGTH & & COVERAGE & LENGTH \\ \cmidrule{2-3}\cmidrule{5-6} $N=1000$ & 0.970 & 2.275 & & 0.967 & 2.837 \\ $N=2500$ & 0.967 & 1.973 & & 0.971 & 2.527 \\ $N=5000$ & 0.954 & 1.743 & & 0.967 & 2.256 \\ \bottomrule \end{tabular} \end{table} \begin{table}[H] \caption{Design 3, Performance of LAD Estimators} \begin{tabular}{lccccccccc} \toprule & \multicolumn{4}{c}{$\beta$} & & \multicolumn{4}{c}{$\gamma$} \\ \cmidrule{2-5}\cmidrule{7-10} & MBIAS & RMSE & MED & MAD & & MBIAS & RMSE & MED & MAD \\ \cmidrule{2-5}\cmidrule{7-10} $N=1000$ & 0.046 & 0.375 & 0.036 & 0.250 & & 0.131 & 0.740 & 0.021 & 0.466 \\ $N=2500$ & 0.026 & 0.205 & 0.020 & 0.133 & & 0.069 & 0.440 & 0.027 & 0.281 \\ $N=5000$ & 0.002 & 0.155 & -0.003 & 0.107 & & 0.067 & 0.327 & 0.039 & 0.214 \\ \midrule & & & & & & & & & \\ \midrule & \multicolumn{4}{c}{$\rho_1$} & & \multicolumn{4}{c}{$\rho_2$} \\ \cmidrule{2-5}\cmidrule{7-10} & MBIAS & RMSE & MED & MAD & & MBIAS & RMSE & MED & MAD \\ \cmidrule{2-5}\cmidrule{7-10} $N=1000$ & 0.027 & 0.320 & -0.006 & 0.198 & & 0.031 & 0.318 & 0.000 & 0.203 \\ $N=2500$ & 0.012 & 0.172 & 0.011 & 0.112 & & 0.009 & 0.167 & 0.012 & 0.114 \\ $N=5000$ & 0.002 & 0.118 & -0.002 & 0.080 & & 0.000 & 0.126 & -0.006 & 0.086 \\ \bottomrule \end{tabular} \end{table} \begin{table}[H] \caption{Design 4, Performance of MS Estimators} \begin{tabular}{lccccccccc} \toprule & \multicolumn{4}{c}{$\beta$} & & \multicolumn{4}{c}{$\gamma$} \\ \cmidrule{2-5}\cmidrule{7-10} & MBIAS & RMSE & MED & MAD & & MBIAS & RMSE & MED & MAD \\ \cmidrule{2-5}\cmidrule{7-10} $N=1000$ & -0.014 & 0.353 & -0.012 & 0.241 & & -0.003 & 0.411 & -0.034 & 0.280 \\ $N=2500$ & 0.000 & 0.318 & -0.007 & 0.236 & & 0.014 & 0.385 & -0.006 & 0.260 \\ $N=5000$ & 0.016 & 0.274 & 0.017 & 0.232 & & 0.027 & 0.354 & 0.002 & 0.246 \\ \bottomrule \end{tabular} \end{table} \begin{table}[H] \caption{Design 4, Performance of Numerical Bootstrap} \begin{tabular}{lccccc} \toprule & \multicolumn{2}{c}{$\beta$} & & \multicolumn{2}{c}{$\gamma$} \\ \cmidrule{2-3}\cmidrule{5-6} & COVERAGE & LENGTH & & COVERAGE & LENGTH \\ \cmidrule{2-3}\cmidrule{5-6} $N=1000$ & 0.976 & 2.159 & & 0.972 & 2.720 \\ $N=2500$ & 0.959 & 1.856 & & 0.969 & 2.383 \\ $N=5000$ & 0.956 & 1.642 & & 0.962 & 2.115 \\ \bottomrule \end{tabular} \end{table} \begin{table}[H] \caption{Design 4, Performance of LAD Estimators} \begin{tabular}{lccccccccc} \toprule & \multicolumn{4}{c}{$\beta$} & & \multicolumn{4}{c}{$\gamma$} \\ \cmidrule{2-5}\cmidrule{7-10} & MBIAS & RMSE & MED & MAD & & MBIAS & RMSE & MED & MAD \\ \cmidrule{2-5}\cmidrule{7-10} $N=1000$ & 0.060 & 0.337 & 0.045 & 0.215 & & 0.132 & 0.666 & 0.025 & 0.406 \\ $N=2500$ & 0.011 & 0.172 & 0.000 & 0.113 & & 0.068 & 0.404 & 0.026 & 0.246 \\ $N=5000$ & 0.002 & 0.128 & -0.001 & 0.087 & & 0.052 & 0.340 & 0.006 & 0.209 \\ \midrule & & & & & & & & & \\ \midrule & \multicolumn{4}{c}{$\rho_1$} & & \multicolumn{4}{c}{$\rho_2$} \\ \cmidrule{2-5}\cmidrule{7-10} & MBIAS & RMSE & MED & MAD & & MBIAS & RMSE & MED & MAD \\ \cmidrule{2-5}\cmidrule{7-10} $N=1000$ & 0.069 & 0.328 & 0.029 & 0.197 & & 0.044 & 0.314 & 0.011 & 0.204 \\ $N=2500$ & -0.015 & 0.163 & -0.024 & 0.110 & & -0.009 & 0.167 & -0.019 & 0.116 \\ $N=5000$ & -0.028 & 0.124 & -0.035 & 0.087 & & -0.024 & 0.128 & -0.024 & 0.087 \\ \bottomrule \end{tabular} \end{table} \section{Proofs for the Cross-Sectional Model} \subsection{Proof of Theorem (ref)} \begin{proof} [Proof of Theorem (ref)]It suffices to show the identification of $\beta $ based on ((ref)) as the same arguments can be applied to the identification of $\beta $ and $ \gamma $ based on similar moment inequalities. Denote $\Omega _{im}=\{X_{i2}=X_{m2},W_{i}=W_{m}\}$. By Assumption C1, the monotonic relation ((ref)) implies that for all $d\in \mathcal{D}$ with $d_{1}=1$, $\beta $ maximizes \begin{equation*} Q_{1}(b)\equiv \mathbb{E} [(P(Y_{i(1,d_{2})}=1|Z_{i})-P(Y_{m(1,d_{2})}=1|Z_{m}))sgn (X_{im1}^{\prime }b)|\Omega _{im}] \end{equation*} for each pair of $(i,m)$. To show that $\beta $ attains a unique maximum, suppose that there is a $b\in \mathcal{B}$ such that $Q_{1}(b)=Q_{1}(\beta )$ . In what follows, we show that $b=\beta $ must hold. To this end, we write $P[ \text{sgn}(X_{im1}^{\prime }b)\neq \text{sgn}(X_{im1}^{\prime }\beta )|\Omega _{im}]=P[(\tilde{X}_{im1}^{\prime }\beta<-X_{im1}^{(1)}<\tilde{X}_{im1}^{\prime }b)\cup ( \tilde{X}_{im1}^{\prime }b<-X_{im1}^{(1)}<\tilde{X} _{im1}^{\prime }\beta)|\Omega _{im}]$. Note that when $b\neq\beta$, $P(\tilde{X} _{im1}^{\prime }\beta\neq\tilde{X}_{im1}^{\prime }b|\Omega _{im})>0$ under Assumption C2(ii). Then by Assumption C2(i), we have $P[(\tilde{X}_{im1}^{\prime }\beta<-X_{im1}^{(1)}<\tilde{X}_{im1}^{\prime }b)\cup ( \tilde{X}_{im1}^{\prime }b<-X_{im1}^{(1)}<\tilde{X}_{im1}^{\prime }\beta)|\Omega _{im}]>0$. Combining with ((ref)), this result implies that the event of $X_{im1}^{\prime }b$ and $P(Y_{i(1,d_{2})}=1|Z_{i})-P(Y_{m(1,d_{2})}=1|Z_{m})$ having different signs occurs with strictly positive probability, and hence $Q_{1}(b)<Q_{1}(\beta )$, a contradiction. Since $b$ is arbitrary, we conclude that $Q_{1}(b)\leq Q_{1}(\beta )$ and the equality holds only when $b=\beta$. This completes the proof. \end{proof} \subsection{Proof of Theorem (ref)} The proof of Theorem (ref) involves utilizing several technical lemmas (i.e., Lemmas (ref)--(ref)). We will present these lemmas prior to the main proof. Before delving into the technical details, we will first introduce some new notations, state Assumption C5 that were omitted from the main text, and provide an outline of the proof process. In what follows, we assume that all regressors are continuous to streamline the exposition.\footnote{This is a purely technical assumption, which is made to facilitate the derivation of the asymptotic variance of the proposed estimators. With discrete covariates, the asymptotic results summarized in Theorem (ref) still valid but with more complex expressions.} The following notations will be used for convenience: For all $d\in\mathcal{D}$ and $j=1,2$, \begin{itemize} • $Y_{imd}\equiv Y_{id}-Y_{md}$. • $f_{X_{imj},W_{im}}(\cdot )$ and $f_{V_{im}\left( \beta \right) }(\cdot )$ denote the PDF of the random vectors $(X_{imj},W_{im})$ and $V_{im}(\beta )$, respectively. $ f_{X_{2},W|X_{1}}(\cdot )$ ($f_{X_{1},W|X_{2}}(\cdot )$) denotes the PDF of $ (X_{2},W)$ ($(X_{1},W)$) conditional on $X_{1}$ ($X_{2}$). $f_{V(\beta )}(\cdot )$ ($ f_{V(\beta )|W}(\cdot )$) is the PDF of $V(\beta )$ (conditional on $W$ ). • For any function $g,\ \nabla g(v)$ and $\nabla ^{2}g(v)$ denote the gradient and Hessian matrix for $g(\cdot )$ evaluated at $v$, respectively. • $V_{i}\equiv V_{i}(\beta)$, $\hat{V}_{i}\equiv V_{i}(\hat{\beta})$, $V_{im}\equiv V_{i}-V_{m}$, and $\hat{V}_{im}\equiv\hat{V}_{i}-\hat{V}_{m}$. • For realized values, $v_{i}\equiv v_{i}(\beta)$, $\hat{v}_{i}\equiv v_{i}(\hat{\beta})$, $v_{im}\equiv v_{i}-v_{m}$, and $\hat{v}_{im}\equiv \hat{v}_{i}-\hat{v}_{m}$. • $B(v_{i},v_{m},w_{i},w_{m})\equiv\mathbb{E} [Y_{im(1,1)}|V_{i}=v_{i},V_{m}=v_{m},W_{i}=w_{i},W_{m}=w_{m}]$. • $S_{im}(r)\equiv \text{sgn}(w_{im}^{\prime }r)-\text{sgn} (w_{im}^{\prime }\gamma )$. • Define \begin{equation} \varrho _{i}(b)\equiv -\sum_{d\in \mathcal{D}}\{\varrho _{i21d}(b)\cdot (-1)^{d_{1}}+\varrho _{i12d}(b)\cdot (-1)^{d_{2}}\}, \end{equation} where \begin{equation*} \varrho _{ijld}(b)\equiv \mathbb{E}\left[ Y_{imd}\left[ sgn (X_{iml}^{\prime }b)-sgn(X_{iml}^{\prime }\beta )\right] |Z_{i},X_{mj}=X_{ij},W_{m}=W_{i}\right] \end{equation*} for $j,l=1,2$ with $j\neq l$. • Define \begin{equation} \tau _{i}(r) \equiv \mathbb{E}[Y_{im(1,1)}\left[ sgn(W_{im}^{\prime }r)-sgn(W_{im}^{\prime }\gamma )\right] |V_{m}\left( \beta \right) =V_{i}\left( \beta \right) ,W_{i}]. \end{equation} • Define \begin{align} &\mu (v_{1},v_{2},r) \\ \equiv &\mathbb{E}\left[ \left. Y_{im(1,1)}\left[ sgn(W_{im}^{\prime }r)-\text{sgn}(W_{im}^{\prime }\gamma )\right] \left( \begin{array}{c} X_{im1}^{\prime } \\ X_{im2}^{\prime } \end{array} \right) \right\vert V_{i}\left( \beta \right) =v_{1},V_{m}\left( \beta \right) =v_{2}\right] f_{V\left( \beta \right) }\left( v_{1}\right). \nonumber \end{align} Let $\nabla _{1}\mu (v_{1},v_{2},r)$ denote the partial derivative of $\mu (v_{1},v_{2},r)$ w.r.t. its first argument. Denote $\nabla _{1k}^{2}\mu (v_{1},v_{2},r)$ as the partial derivative of $\nabla _{1}\mu (v_{1},v_{2},r)$ w.r.t. its $k$-th argument and $\nabla _{33}^{2}\nabla _{1}\mu (v_{1},v_{2},r)$ as the Hessian matrix of $\nabla _{1}\mu (v_{1},v_{2},r)$ w.r.t. its third argument. • $K_{h_{N},\beta}(\cdot)\equiv h_{N}^{k_{1}+k_{2}}\mathcal{K}_{h_{N}}(\cdot )$ and $K_{\sigma_{N},\gamma}(\cdot)\equiv \sigma _{N}^{2}\mathcal{K}_{\sigma _{N}}(\cdot )$. In proofs, we also write $K_{\beta}(\cdot/h_{N})$ and $K_{\gamma}(\cdot/\sigma_{N})$ for $K_{h_{N},\beta}(\cdot)$ and $K_{\sigma_{N},\gamma}(\cdot)$, respectively, when we need to apply changes of variables. \end{itemize} The following regularity conditions that were omitted in the main text will be used in proving Lemmas (ref)--(ref) and Theorem (ref): \begin{itemize} • (i) $f_{V_{im}\left( \beta \right) }(\cdot )$ and $ f_{X_{imj},W_{im}}(\cdot )$ for $j=1,2$ are bounded from above on their supports, strictly positive in a neighborhood of zero, and twice continuously differentiable with bounded second derivatives, (ii) For all $ d\in \mathcal{D}$, $b\in \mathcal{B}$, and $r\in \mathcal{R}$, $\mathbb{E} [Y_{imd}[\text{sgn}(X_{im1}^{\prime }b)-\text{sgn}(X_{im1}^{\prime }\beta )]|X_{im2}=\cdot ,W_{im}=\cdot ]$, $\mathbb{E}[Y_{imd}[\text{sgn} (X_{im2}^{\prime }b)-\text{sgn}(X_{im2}^{\prime }\beta )]|X_{im1}=\cdot ,W_{im}=\cdot ]$, and $\mathbb{E}[Y_{im(1,1)}[\text{sgn}(W_{im}^{\prime }r)- \text{sgn}(W_{im}^{\prime }\gamma )]|V_{im}\left( \beta \right) =\cdot ]$ are continuously differentiable with bounded first derivatives, (iii) $ \mathbb{E}[Y_{imd}|Z_{i}=\cdot ,Z_{m}=\cdot ]$ is $\kappa _{1\beta }^{\text{ th}}$ continuously differentiable with bounded $\kappa _{1\beta }^{\text{th} } $ derivatives. $f_{X_{2},W|X_{1}}(\cdot )$ ($f_{X_{1},W|X_{2}}(\cdot )$) is $\kappa _{2\beta }^{\text{th}}$ continuously differentiable with bounded $ \kappa _{2\beta }^{\text{th}}$ derivatives. Denote $\kappa _{\beta }=\kappa _{1\beta }+\kappa _{2\beta }$. $\kappa _{\beta }$ is an even integer greater than $k_{1}+k_{2}$, (iv) $\mathbb{E}[Y_{im(1,1)}|V_{i}=\cdot ,V_{m}=\cdot ,W_{i}=\cdot ,W_{m}=\cdot ]$ is $\kappa _{1\gamma }^{\text{th}}$ continuously differentiable with bounded $\kappa _{1\gamma }^{\text{th}}$ derivatives, and $f_{V(\beta )|W}(\cdot )$ is $\kappa _{2\gamma }^{\text{th} } $ continuously differentiable with bounded $\kappa _{2\gamma }^{\text{th}}$ derivatives. Denote $\kappa _{\gamma }=\kappa _{1\gamma }+\kappa _{2\gamma }$ . $\kappa _{\gamma }$ is an even integer greater than 2, and (v) $\mathbb{E} [\Vert X_{imj}\Vert ^{2}]<\infty $ for all $j=1,2,$ and $\mathbb{E}[\Vert W_{im}\Vert ^{2}]<\infty $. All derivatives in this assumption are with respect to $\cdot $. \end{itemize} The boundedness and smoothness restrictions under Assumption C5 are needed for proving the uniform convergence of the criterion functions to their population analogues. \begin{comment} Before the main proofs, we first present supporting lemmas, Lemmas (ref) -- (ref), for the proof of Theorem (ref). The proofs of the support lemmas are relegated to Appendix (ref). \end{comment} In what follows, we will only show the asymptotics of $\hat{\gamma}$. The proof for $\hat {\beta}$ is omitted since one can derive the asymptotics of $ \hat{\beta}$ by repeating the proof process for $\hat{\gamma}$ but skipping the step handling the plug-in first step estimates (i.e., Lemma (ref) ). Throughout this section, we take it as a given that $\hat{\beta}$ is $ \sqrt{N}$-consistent and asymptotically normal. \begin{comment} \textbf{To obtain }$\psi_{i,\beta}$\textbf{, repeat the entire process for }$\hat{\beta}$\textbf{ but the omit the part for the effects of the first step estimator. Specifically, }$\psi_{i,\beta}$ is the analogous part of $\varPsi_{N,1}$ for $\hat{\gamma}$\textbf{. Then} \[ \psi_{i,\beta}= \] \textbf{Thus} \[ \sqrt{N}\left( \hat{\beta}-\beta\right) \overset{d}{\rightarrow}. \] \end{comment} For ease of illustration, with a bit of abuse of notation, we will work with criterion function \begin{equation*} \hat{\mathcal{L}}_{N}^{K}(r)\equiv\frac{1}{\sigma_{N}^{2}N(N-1)}\sum_{i\neq m}K_{\sigma_{N},\gamma}(V_{im}(\hat{\beta}))h_{im}(r), \end{equation*} where $K_{\sigma_{N},\gamma}(\cdot)\equiv\sigma_{N}^{2}\mathcal{K}_{\sigma _{N}}(\cdot)$ and $h_{im}(r)\equiv Y_{im(1,1)}[\text{sgn}(W_{im}^{\prime }r)- \text{sgn}(W_{im}^{\prime}\gamma)]$. Note that here we subtract the term $\text{sgn}(W_{im}^{\prime}\gamma)$ from criterion function ((ref)), analogous to Sherman1993. Doing this does not affect the value of the estimator, and will facilitate the proofs that follow. Besides, we define \begin{equation*} \mathcal{L}_{N}^{K}(r)=\frac{1}{\sigma_{N}^{2}N(N-1)}\sum_{i\neq m}K_{\sigma_{N},\gamma}(V_{im}(\beta))h_{im}(r) \end{equation*} and \begin{equation*} \mathcal{L}(r)\equiv f_{V_{im}(\beta)}(0)\mathbb{E}[h_{im}(r)|V_{im}( \beta)=0]. \end{equation*} We establish the consistency of $\hat{\gamma }$ in Lemmas (ref)--(ref) by applying Theorem 2.1 in NeweyMcFadden1994. The key step is to show the uniform convergence of $\hat{\mathcal{L}}_{N}^{K}(r)$ for $ r\in\mathcal{R}$. To this end, we bound the differences among $\hat { \mathcal{L}}_{N}^{K}(r)$, $\mathcal{L}_{N}^{K}(r)$, and $\mathcal{L}(r)$ in Lemmas (ref) and (ref). The next step is to show the asymptotic normality of $\hat{\gamma}$ by applying Theorem 2 of Sherman1994AoS. Sufficient conditions for this theorem are that $\hat{\gamma}-\gamma=O_{p}(N^{-1/2})$ and uniformly over a neighborhood of $\gamma$ with a radius proportional to $N^{-1/2}$, \begin{equation} \hat{\mathcal{L}}_{N}^{K}(r)=\frac{1}{2}(r-\gamma)^{\prime}\mathbb{V} _{\gamma }(r-\gamma)+\frac{1}{\sqrt{N}}(r-\gamma)^{\prime}\varPsi _{N}+o_{p}(N^{-1}), \end{equation} where $\mathbb{V}_{\gamma}$ is a negative definite matrix and $\varPsi_{N}$ is asymptotically normal, with mean zero and variance $\mathbb{V}_{\varPsi}$ . To verify equation ((ref)), we first show that uniformly over a neighborhood of $\gamma$, $\mathcal{R}_{N}\equiv\{r\in\mathcal{R}|\Vert r-\gamma\Vert\leq\delta_{N}\}$ with $\{\delta_{N}\}=O(N^{-\delta})$ for some $0<\delta\leq1/2$, \begin{equation} \hat{\mathcal{L}}_{N}^{K}(r)=\frac{1}{2}(r-\gamma)^{\prime}\mathbb{V} _{\gamma }(r-\gamma)+\frac{1}{\sqrt{N}}(r-\gamma)^{\prime}\varPsi _{N}+o_{p}(\Vert r-\gamma\Vert^{2})+O_{p}(\varepsilon_{N}), \end{equation} which is the task of Lemmas (ref)--(ref). The $ O_{p}(\varepsilon_{N})$ term in equation ((ref)) can be shown to be of order $o_{p}(N^{-1})$ uniformly over $\mathcal{R}_{N}$. Further, by Theorem 1 of Sherman1994ET, equation ((ref)) implies $\hat{\gamma} -\gamma=O_{p}(\sqrt{\varepsilon_{N}}\vee N^{-1/2})=O_{p}(N^{-1/2})$. This rate result, together with equation ((ref)), further verifies equation ((ref)). To obtain equation ((ref)), we will work with the following expansion \begin{align} \hat{\mathcal{L}}_{N}^{K}(r) & =\mathcal{L}_{N}^{K}(r)+\Delta\mathcal{L} _{N}^{K}(r)+R_{N} \notag \\ & =\mathbb{E}[\mathcal{L}_{N}^{K}(r)]+\frac{2}{N}\sum_{i}\left\{ \mathbb{E}[ \mathcal{L}_{N}^{K}(r)|Z_{i}]-\mathbb{E}[\mathcal{L}_{N}^{K}(r)]\right\} +\rho_{N}(r)+\Delta\mathcal{L}_{N}^{K}(r)+R_{N}, \end{align} where \begin{equation} \Delta\mathcal{L}_{N}^{K}(r)\equiv\frac{1}{\sigma_{N}^{3}N(N-1)}\sum_{i\neq m}\nabla K_{\sigma_{N},\gamma}(V_{im}(\beta))^{\prime}(V_{im}(\hat{\beta } )-V_{im}\left( \beta\right) )h_{im}(r), \end{equation} \begin{equation*} \rho_{N}(r)=\mathcal{L}_{N}^{K}(r)-\frac{2}{N}\sum_{i}\mathbb{E}[\mathcal{L} _{N}^{K}(r)|Z_{i}]+\mathbb{E}[\mathcal{L}_{N}^{K}(r)], \end{equation*} and $R_{N}$ denotes the remainder term in the expansion of higher order (as $ \sqrt{N}\sigma_{N}\rightarrow\infty$). The first three terms in ((ref) ) are the U-statistic decomposition for $\mathcal{L}_{N}^{K}(r)$ (see e.g., Sherman1993 and Serfling2009). Lemmas (ref)--(ref) establish asymptotic properties of these three terms, respectively. A linear representation for the fourth term in ((ref)) is derived in Lemma (ref). Here we present Lemmas (ref)--(ref), whose proofs are relegated to Appendix \hyperref[appendixE]{F}. \begin{lemma} Under Assumptions C1--C7, $\sup_{r\in \mathcal{R}}|\hat{ \mathcal{L}}_{N}^{K}(r)-\mathcal{L}_{N}^{K}(r)|=o_{p}(1)$. \end{lemma} \begin{lemma} Under Assumptions C1 and C5--C7, $\sup_{r\in \mathcal{R} }|\mathcal{L}_{N}^{K}(r)-\mathcal{L}(r)|=o_{p}(1)$. \end{lemma} \begin{lemma} Under Assumptions C1--C7,, $\hat{\gamma}\overset{p}{ \rightarrow }\gamma $. \end{lemma} \begin{lemma} Under Assumptions C1--C7,, uniformly over $\mathcal{R}_{N}$, we have \begin{equation*} \mathbb{E}[\mathcal{L}_{N}^{K}(r)]=\frac{1}{2}(r-\gamma )^{\prime }\mathbb{V} _{\gamma }(r-\gamma )+o(\Vert r-\gamma \Vert ^{2}), \end{equation*} where $\mathbb{V}_{\gamma }\equiv \mathbb{E}[\nabla ^{2}\tau _{i}(\gamma )]$. \end{lemma} \begin{lemma} Under Assumptions C1--C7,, uniformly over $\mathcal{R}_{N}$, we have \begin{equation*} \frac{2}{N}\sum_{m}\mathbb{E}[\mathcal{L}_{N}^{K}(r)|Z_{m}]-2\mathbb{E}[ \mathcal{L}_{N}^{K}(r)]=\frac{1}{\sqrt{N}}(r-\gamma )^{\prime }\varPsi _{N,1}+o_{p}(\Vert r-\gamma \Vert ^{2}), \end{equation*} where $\varPsi_{N,1}\equiv N^{-1/2}\sum_{m}2\nabla \tau _{m}(\gamma )$. \end{lemma} \begin{lemma} Under Assumptions C1 and C6--C7, uniformly over $\mathcal{R} _{N}$, $\rho _{N}(r)=O_{p}(N^{-1}\sigma _{N}^{-2})$. \end{lemma} \begin{lemma} Suppose \begin{equation} \hat{\beta}-\beta=\frac{1}{N}\sum_{i}\psi_{i,\beta}+o_{p}(N^{-1/2}), \end{equation} where $\psi_{i,\beta}$ is the influence function (of $Z_{i}$). $ \psi_{i,\beta }$ is i.i.d across $i$ with $\mathbb{E}\left( \psi_{i,\beta}\right) =0$.\footnote{ In fact, the linear representation of $\hat{\beta}-\beta$ in ((ref)) can be obtained by applying the same arguments in Theorem 2 of Sherman1993 to a representation for $\mathcal{L}_{N,\beta}^{K}(b)$ analogous to ((ref)).} Further, suppose Assumptions C1--C7 hold. Then uniformly over $\mathcal{R}_{N}$, we have \begin{equation*} \Delta\mathcal{L}_{N}^{K}(r)=\frac{1}{\sqrt{N}}(r-\gamma)^{\prime }\varPsi _{N,2}+o_{p}(\Vert r-\gamma\Vert^{2})+O_{p}(N^{-1}\sigma_{N}^{-2}), \end{equation*} with $\varPsi_{N,2}=\frac{1}{\sqrt{N}}\sum_{i}\left( -\int\nabla_{13}^{2}\mu(v_{m},v_{m},\gamma)f_{V}(v_{m})dv_{m}\right) \psi_{i,\beta}$, where $\mu(v_{i},v_{m},r)\equiv G(v_{i},v_{m},r)f_{V}(v_{i}) $, \begin{equation*} G(v_{i},v_{m},r)\equiv\mathbb{E}\left[ B(V_{i},V_{m},W_{i},W_{m})S_{im}(r) \left( \begin{array}{c} x_{im1}^{\prime} \\ x_{im2}^{\prime} \end{array} \right) |V_{i}=v_{i},V_{m}=v_{m}\right] , \end{equation*} $\nabla_{1}\mu(\cdot,\cdot,\cdot)$ denotes the partial derivative of $ \mu(\cdot,\cdot,\cdot)$ w.r.t. its first argument, and $\nabla_{13}^{2}\mu( \cdot,\cdot,\cdot)$ denotes the partial derivative of $\nabla_{1}\mu (\cdot,\cdot,\cdot)$ w.r.t. its third argument. \end{lemma} \begin{proof}[Proof of Theorem (ref)] Putting results in Lemmas (ref)--(ref) together, we write equation ((ref)) as \begin{equation} \hat{\mathcal{L}}_{N}^{K}(r)=\frac{1}{2}(r-\gamma)^{\prime}\mathbb{V} _{\gamma }(r-\gamma)+\frac{1}{\sqrt{N}}(r-\gamma)^{\prime}\varPsi _{N}+o_{p}(\Vert r-\gamma\Vert^{2})+O_{p}(\varepsilon_{N}) \end{equation} where $\varPsi_{N}=\varPsi_{N,1}+\varPsi_{N,2}$ and $\varepsilon_{N}=N^{-1} \sigma_{N}^{-2}$. Theorem 1 of Sherman1994ET then implies that $\hat{ \gamma}-\gamma=O_{p}(\sqrt{\varepsilon_{N}})=O_{p}(N^{-1/2}\sigma _{N}^{-1})$ . Next, take $\delta_{N}=O(\sqrt{\varepsilon_{N}})$ and $\mathcal{R}_{N}=\{r\in \mathcal{R}|\Vert r-\gamma\Vert\leq\delta_{N}\}$. We repeat the proof for Lemma (ref) and deduce from a Taylor expansion around $\gamma$ that $ \sup_{r\in\mathcal{R}_{N}}\mathbb{E}[\rho_{im}^{\ast}(r)^{2}]=O( \sigma_{N}^{2}\delta_{N}^{2})$. Apply Theorem 3 of Sherman1994ET to see that uniformly over $\mathcal{R}_{N}$, $\rho_{N}(r)=O_{p}(N^{-1}\sigma _{N}^{\lambda-2}\delta_{N}^{\lambda})$ where $0<\lambda<1$. Then we have \begin{equation*} \rho_{N}(r)=O_{p}(N^{-1}\sigma_{N}^{\lambda-2}\delta_{N}^{ \lambda})=O_{p}(N^{-1})O_{p}(N^{-\lambda/2}\sigma_{N}^{-2})=o_{p}(N^{-1}) \end{equation*} by invoking Assumption C7 and choosing $\lambda$ sufficiently close to 1. This result in turn implies that the $O_{p}(\varepsilon_{N})$ term in ((ref)) has order $o_{p}(N^{-1})$, and hence $\hat{\gamma}-\gamma =O_{p}(N^{-1/2})$ by applying Theorem 1 of Sherman1994ET once again. Now ((ref)) can be expressed as \begin{equation*} \hat{\mathcal{L}}_{N}^{K}(r)=\frac{1}{2}(r-\gamma)^{\prime}\mathbb{V} _{\gamma }(r-\gamma)+\frac{1}{\sqrt{N}}(r-\gamma)^{\prime}\varPsi _{N}+o_{p}(N^{-1}). \end{equation*} Let $\varDelta_{i}\equiv2\nabla\tau_{i}(\gamma)-\left( \int\nabla_{13}^{2}\mu(v_{m},v_{m},\gamma)f_{V}(v_{m})\text{d}v_{m}\right) \psi_{i,\beta}$. Note that $\mathbb{E}[\varDelta_{i}]=0$ since $\mathbb{E} [\nabla\tau_{i}(\gamma)]=0$ and $\mathbb{E}[\psi_{i,\beta}]=0$. We deduce from Assumption C7 and Lindeberg-L\'{e}vy CLT that $\varPsi_{N}\overset{d}{ \rightarrow }N(0,\mathbb{V}_{\varPsi})$ where $\mathbb{V}_{\varPsi}=\mathbb{E }[\varDelta_{i}\varDelta_{i}^{\prime}]$. The asymptotic normality of $\hat{ \gamma}$ then follows from Theorem 2 of Sherman1994AoS, i.e., $\sqrt{N }(\hat{\gamma}-\gamma)\overset{d}{\rightarrow}N(0,\mathbb{V}_{\gamma }^{-1} \mathbb{V}_{\varPsi}\mathbb{V}_{\gamma}^{-1})$. \end{proof} \subsection{Proof of of Theorem (ref)} Recall that $\mathcal{\hat{L}}_{N}^{K}\left( r\right) $ was redefined in Appendix (ref). However, for the proof of Theorem (ref), it is more convenient to still use the definition in the main body of the paper, specifically, \begin{equation*} \mathcal{\hat{L}}_{N}^{K}\left( \hat{\gamma}\right) =\frac{1}{\sigma _{N}^{2}N\left( N-1\right) }\sum_{i\neq m}K_{\sigma _{N},\gamma } ( V_{im} ( \hat{\beta} ) ) Y_{im\left( 1,1\right) }\text{sgn} \left( W_{im}^{\prime }\hat{\gamma}\right) . \end{equation*} \begin{lemma} Suppose Assumptions C1--C7 hold. Then, we have \begin{equation*} \frac{1}{\sigma _{N}^{2}N\left( N-1\right) }\sum_{i\neq m} [ K_{\sigma _{N},\gamma } ( V_{im} ( \hat{\beta} ) ) -K_{\sigma _{N},\gamma }\left( V_{im}\left( \beta \right) \right) ] Y_{im\left( 1,1\right) }\text{\emph{sgn}}\left( W_{im}^{\prime }\gamma \right) =O_{p} ( N^{-1}\sigma _{N}^{-2} ) . \end{equation*} \end{lemma} \begin{lemma} Suppose Assumptions C1--C7 hold. Then uniformly over $r\in \mathcal{R}$, \begin{eqnarray*} \mathcal{\hat{L}}_{N}^{K}\left( r\right) &=&\frac{1}{2}\left( r-\gamma \right) ^{\prime }\mathbb{V}_{\gamma }\left( r-\gamma \right) +o_{p}\left( N^{-1/2}\left\Vert r-\gamma \right\Vert \right) +o_{p}\left( \left\Vert r-\gamma \right\Vert ^{2}\right) \\ &&+\frac{1}{\sigma _{N}^{2}N\left( N-1\right) }\sum_{i\neq m}K_{\sigma _{N},\gamma }\left( V_{im}\left( \beta \right) \right) Y_{im\left( 1,1\right) }\textrm{sgn}\left( W_{im}^{\prime }\gamma \right) +o_{p}( N^{-1/2}) . \end{eqnarray*} \end{lemma} \begin{lemma} Suppose Assumptions C5--C7 hold. Then, we have \begin{eqnarray*} &&\mathbb{E}\left[ \sigma _{N}^{-2}K_{\sigma _{N},\gamma }\left( V_{im}\left( \beta \right) \right) Y_{im\left( 1,1\right) }\textrm{sgn}\left( W_{im}^{\prime }\gamma \right) |Z_{i}\right] \\ &=&f_{V_{im}(\beta)}(0)\mathbb{E}\left[ Y_{im\left( 1,1\right) }\textrm{sgn}\left( W_{im}^{\prime }\gamma \right) |Z_{i},V_{im}\left( \beta \right) =0 \right] +o_{p}( N^{-1/2}) . \end{eqnarray*} \end{lemma} Note that Lemma (ref) indicates that $\hat{\beta}$ affects $\mathcal{\hat{ L}}_{N}^{K}\left( \hat{\gamma}\right) $ at the rate of $N^{-1}\sigma _{N}^{-2}$, and Lemma (ref) shows that $\hat{\gamma}$ affects $ \mathcal{\hat{L}}_{N}^{K}\left( \hat{\gamma}\right) $ at the rate of $ N^{-1}$. \begin{proof} [Proof of Theorem (ref)] Note that $O_p(N^{-1}\sigma_N^{-2})=o_p(N^{-1/2})$ by Assumption C7 and $\hat{\gamma}-$ $\gamma =O_{p}( N^{-1/2})$ by Theorem (ref). These results, together with Lemmas (ref) and (ref), imply that \begin{equation*} \mathcal{\hat{L}}_{N}^{K}\left( \hat{\gamma}\right) =\frac{1}{\sigma _{N}^{2}N\left( N-1\right) }\sum_{i\neq m}K_{\sigma _{N},\gamma }\left( V_{im}\left( \beta \right) \right) Y_{im\left( 1,1\right) }\text{sgn}\left( W_{im}^{\prime }\gamma \right) +o_{p} ( N^{-1/2} ) . \end{equation*} We can use the same arguments as those used for proving Lemmas (ref) and (ref) to show \begin{equation*} \mathcal{\hat{L}}_{N}^{K}\left( \hat{\gamma}\right) \overset{p}{\rightarrow } \mathcal{\bar{L}}\left( \gamma \right) \mathcal{=}f_{V_{im}\left( \beta \right) }\left( 0\right) \mathbb{E}\left[ Y_{im\left( 1,1\right) }\text{sgn} \left( W_{im}^{\prime }\gamma \right) |V_{im}\left( \beta \right) =0\right]. \end{equation*} Then, we can write \begin{eqnarray*} &&\mathcal{\hat{L}}_{N}^{K}\left( \hat{\gamma}\right) -\mathcal{\bar{L}} \left( \gamma \right) \\ &=&\frac{1}{\sigma _{N}^{2}N\left( N-1\right) } \sum_{i\neq m}K_{\sigma _{N},\gamma }\left( V_{im}\left( \beta \right) \right) Y_{im\left( 1,1\right) }\text{sgn}\left( W_{im}^{\prime }\gamma \right) -\mathcal{\bar{L}}\left( \gamma \right) +o_{p}( N^{-1/2}) \\ &=&\frac{2}{N}\sum_{i=1}^{N}\left\{ \mathbb{E}\left[ \sigma _{N}^{-2}K_{\sigma _{N},\gamma }\left( V_{im}\left( \beta \right) \right) Y_{im\left( 1,1\right) }\text{sgn}\left( W_{im}^{\prime }\gamma \right) |Z_{i}\right] -\mathbb{E}\left[ \sigma _{N}^{-2}K_{\sigma _{N},\gamma }\left( V_{im}\left( \beta \right) \right) Y_{im\left( 1,1\right) }\text{sgn} \left( W_{im}^{\prime }\gamma \right) \right] \right\} \\ &&+\mathbb{E}\left[ \sigma _{N}^{-2}K_{\sigma _{N},\gamma }\left( V_{im}\left( \beta \right) \right) Y_{im\left( 1,1\right) }\text{sgn}\left( W_{im}^{\prime }\gamma \right) \right] -\mathcal{\bar{L}}\left( \gamma \right) +o_{p} ( N^{-1/2} ) \\ &=&\frac{2}{N}\sum_{i=1}^{N}f_{V_{im}(\beta)}(0)\left\{ \mathbb{E}\left[ Y_{im\left( 1,1\right) } \text{sgn}\left( W_{im}^{\prime }\gamma \right) |Z_{i},V_{im}\left( \beta \right) =0\right] -\mathbb{E}\left[ Y_{im\left( 1,1\right) }\text{sgn}\left( W_{im}^{\prime }\gamma \right) |V_{im}\left( \beta \right) =0\right] \right\} \\ &&+o_{p}( N^{-1/2}), \end{eqnarray*} where the second equality uses the decomposition ((ref)) and Lemma (ref), and the last equality follows by Lemma (ref). Invoking the Lindeberg-L\'{e}vy CLT on the leading term of $\mathcal{\hat{L}} _{N}^{K}\left( \hat{\gamma}\right) -\mathcal{\bar{L}(\gamma)}$ in the above yields \begin{equation*} \sqrt{N}( \mathcal{\hat{L}}_{N}^{K}\left( \hat{\gamma}\right) -\mathcal{ \bar{L}}\left( \gamma \right) ) \overset{d}{\rightarrow }N\left( 0,\Delta_\gamma \right) . \end{equation*} \end{proof} \subsection{Proof of of Theorem (ref)} \begin{proof}[Proof of Theorem (ref)] The proof is lengthy, but the underlying idea is straightforward. Using the (conditional) linear independence (equivalent to the regular full rank condition) and the free variation of the first regressor conditions in Assumption CL2, we show that there exists a positive probability of making an incorrect prediction for any $\vartheta\neq\theta$. \textbf{Some Definitions.} We normalize the coefficients on $X_{im1}^{(1)}$, $X_{im2}^{(1)}$, and $W_{i}^{(1)}$ to 1 as indicated in Assumption CL3. To simplify notation, we express the three utility indexes with a slight abuse of notation as $X_{im1}^{(1)}+u_{im1}(\vartheta)$, $X_{im2}^{(1)}+u_{im2}(\vartheta)$, and $W_{im}^{(1)}+u_{imb}(\vartheta)$, where $u_{im1}(\vartheta)\equiv\tilde{X}_{im1}^{\prime}b+S_{im}^{\prime }\varrho_{1}$, $u_{im2}(\vartheta)\equiv\tilde{X}_{im2}^{\prime} b+S_{im}^{\prime}\varrho_{2}$, and $u_{imb}(\vartheta)\equiv\tilde{W} _{im}^{\prime}r+S_{im}^{\prime}\varrho_{b}$, respectively. We use $a\vee b$ and $a\wedge b$ to denote $\max\{a,b\}$ and $\min\{a,b\}$, respectively. Finally, we define indicator functions for the events on the left-hand sides of ((ref))--((ref)), ((ref))--((ref)), and ((ref))--((ref))\footnote{There is no need to define $I_{im(0,0)}^{+}(\vartheta)$ and $I_{im(0,0)}^{-}(\vartheta)$ separately since $I_{im(0,0)}^{+}(\vartheta)=I_{im(1,1)}^{-}(\vartheta)$ and $I_{im(0,0)}^{-}(\vartheta)=I_{im(1,1)}^{+}(\vartheta)$.} and the two uninformative events in Remark (ref) in turn: \begin{align*} I_{im(0,1)}^{+}(\vartheta) & =1[X_{im1}^{(1)}+u_{im1}(\vartheta)\leq 0,X_{im2}^{(1)}+u_{im2}(\vartheta)\geq0,W_{im}^{(1)}+u_{imb}(\vartheta )\leq0],\\ I_{im(0,1)}^{-}(\vartheta) & =1[X_{im1}^{(1)}+u_{im1}(\vartheta)\geq 0,X_{im2}^{(1)}+u_{im2}(\vartheta)\leq0,W_{im}^{(1)}+u_{imb}(\vartheta )\geq0],\\ I_{im(1,1)}^{+}(\vartheta) & =1[X_{im1}^{(1)}+u_{im1}(\vartheta)\geq 0,X_{im2}^{(1)}+u_{im2}(\vartheta)\geq0,W_{im}^{(1)}+u_{imb}(\vartheta )\geq0],\\ I_{im(1,1)}^{-}(\vartheta) & =1[X_{im1}^{(1)}+u_{im1}(\vartheta)\leq 0,X_{im2}^{(1)}+u_{im2}(\vartheta)\leq0,W_{im}^{(1)}+u_{imb}(\vartheta )\leq0],\\ I_{imu}^{+}(\vartheta) & =1[X_{im1}^{(1)}+u_{im1}(\vartheta)\leq 0,X_{im2}^{(1)}+u_{im2}(\vartheta)\leq0,W_{im}^{(1)}+u_{imb}(\vartheta )\geq0],\\ I_{imu}^{-}(\vartheta) & =1[X_{im1}^{(1)}+u_{im1}(\vartheta)\geq 0,X_{im2}^{(1)}+u_{im2}(\vartheta)\geq0,W_{im}^{(1)}+u_{imb}(\vartheta)\leq0]. \end{align*} Recall that by construction $q_{(1,0)}(Z_{i},Z_{m},\theta)=0\leq q_{(1,0)}(Z_{i},Z_{m},\vartheta)$ for all $\vartheta\in\Theta$ and $(Z_{i},Z_{m})$. In what follows, we show for any $\vartheta\in\Theta \setminus\{\theta\}$, $q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0$ occurs with positive probability, and hence $Q_{(1,0)}(\theta)<Q_{(1,0)}(\vartheta)$. \textbf{Identification of }$(\beta,\rho_{1})$\textbf{.} For any $\vartheta\in\Theta\setminus\{\theta\}$, we have \begin{align} & P(q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0)\nonumber\\ \geq & P(\{q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0\}\cap\{I_{im(1,1)}^{+} (\theta)+I_{im(1,1)}^{-}(\theta)=1\})\nonumber\\ = & P(\{q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0\}\cap\{I_{im(1,1)}^{+} (\theta)=1\})+P(\{q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0\}\cap\{I_{im(1,1)} ^{-}(\theta)=1\})\nonumber\\ \geq & P(\{q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0\}\cap\{I_{im(1,1)}^{+} (\theta)=I_{im(1,0)}^{-}(\vartheta)=1\})\nonumber\\ & +P(\{q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0\}\cap\{I_{im(1,1)}^{-} (\theta)=I_{im(1,0)}^{+}(\vartheta)=1\})\nonumber\\ = & P(q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0|I_{im(1,1)}^{+}(\theta )=I_{im(1,0)}^{-}(\vartheta)=1)P(I_{im(1,1)}^{+}(\theta)=I_{im(1,0)} ^{-}(\vartheta)=1)\nonumber\\ & +P(q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0|I_{im(1,1)}^{-}(\theta)=I_{im(1,0)} ^{+}(\vartheta)=1)P(I_{im(1,1)}^{-}(\theta)=I_{im(1,0)}^{+}(\vartheta )=1). \end{align} By definitions, we can write \begin{align*} & P(I_{im(1,1)}^{+}(\theta)=I_{im(1,0)}^{-}(\vartheta)=1)\\ = & P(u_{im1}(\vartheta)\leq-X_{im1}^{(1)}\leq u_{im1}(\theta),-X_{im2} ^{(1)}\leq u_{im2}(\vartheta)\wedge u_{im2}(\theta),-W_{im}^{(1)}\leq u_{imb}(\vartheta)\wedge u_{imb}(\theta)), \end{align*} and \begin{align*} & P(I_{im(1,1)}^{-}(\theta)=I_{im(1,0)}^{+}(\vartheta)=1)\\ = & P(u_{im1}(\theta)\leq-X_{im1}^{(1)}\leq u_{im1}(\vartheta),-X_{im2} ^{(1)}\geq u_{im2}(\vartheta)\vee u_{im2}(\theta),-W_{im}^{(1)}\geq u_{imb}(\vartheta)\vee u_{imb}(\theta)). \end{align*} Assumptions CL2(i) and CL2(iii) imply that events $\{-X_{im2}^{(1)}\leq u_{im2}(\vartheta)\wedge u_{im2}(\theta),-W_{im}^{(1)}\leq u_{imb} (\vartheta)\wedge u_{imb}(\theta)\}$ and $\{-X_{im2}^{(1)}\geq u_{im2} (\vartheta)\vee u_{im2}(\theta),-W_{im}^{(1)}\geq u_{imb}(\vartheta)\vee u_{imb}(\theta)\}$ always have positive probabilities of occurring. Then, when conditioning on them, Assumption CL2(ii) implies that the event ${u_{im1} (\vartheta)\neq u_{im1}(\theta)}$ has a positive probability to occur when $(b,\varrho_{1})\neq(\beta,\rho_{1})$. Thus, under Assumption CL2(i), we conclude that either ${u_{im1}(\vartheta)\leq-X_{im1}^{(1)}\leq u_{im1} (\theta)}$ or ${u_{im1}(\theta)\leq-X_{im1}^{(1)}\leq u_{im1}(\vartheta)}$ occurs with positive probability. In summary, we can claim that for any $\vartheta$ with $(b,\varrho_{1})\neq(\beta,\rho_{1})$, \begin{equation} P(I_{im(1,1)}^{+}(\theta)=I_{im(1,0)}^{-}(\vartheta)=1)>0\text{ or }P(I_{im(1,1)}^{-}(\theta)=I_{im(1,0)}^{+}(\vartheta)=1)>0. \end{equation} Then, if we can establish that \begin{align} P(q_{(1,0)}(Z_{i},Z_{m},\vartheta) & >0|I_{im(1,1)}^{+}(\theta )=I_{im(1,0)}^{-}(\vartheta)=1)>0\text{ and}\nonumber\\ P(q_{(1,0)}(Z_{i},Z_{m},\vartheta) & >0|I_{im(1,1)}^{-}(\theta )=I_{im(1,0)}^{+}(\vartheta)=1)>0, \end{align} by substituting ((ref)) into ((ref)), we obtain $P(q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0)>0$ for any $(b,\varrho_{1})\neq (\beta,\rho_{1})$. This establishes the identification of $(\beta,\rho_{1})$ in $\theta$. We defer the justification of the inequalities in ((ref)) to the end of this proof. \textbf{Identification of }$(\beta,\rho_{2})$\textbf{.} Similarly, we can write \begin{align} & P(q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0)\nonumber\\ \geq & P(\{q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0\}\cap\{I_{imu}^{+} (\theta)+I_{imu}^{-}(\theta)=1\})\nonumber\\ \geq & P(q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0|I_{imu}^{+}(\theta)=I_{im(1,0)} ^{-}(\vartheta)=1)P(I_{imu}^{+}(\theta)=I_{im(1,0)}^{-}(\vartheta )=1)\nonumber\\ & +P(q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0|I_{imu}^{-}(\theta)=I_{im(1,0)} ^{+}(\vartheta)=1)P(I_{imu}^{-}(\theta)=I_{im(1,0)}^{+}(\vartheta )=1), \end{align} where \begin{align*} & P(I_{imu}^{+}(\theta)=I_{im(1,0)}^{-}(\vartheta)=1)\\ = & P(-X_{im1}^{(1)}\geq u_{im1}(\vartheta)\vee u_{im1}(\theta),u_{im2} (\theta)\leq-X_{im2}^{(1)}\leq u_{im2}(\vartheta),-W_{im}^{(1)}\leq u_{imb}(\vartheta)\wedge u_{imb}(\theta)), \end{align*} and \begin{align*} & P(I_{imu}^{-}(\theta)=I_{im(1,0)}^{+}(\vartheta)=1)\\ = & P(-X_{im1}^{(1)}\leq u_{im1}(\vartheta)\wedge u_{im1}(\theta ),u_{im2}(\vartheta)\leq-X_{im2}^{(1)}\leq u_{im2}(\theta),-W_{im}^{(1)}\geq u_{imb}(\vartheta)\vee u_{imb}(\theta)). \end{align*} Then, we can establish the identification of $(\beta,\rho_{2})$ in $\theta$ using similar arguments as above. \textbf{Identification of }$(\gamma,\rho_{b})$\textbf{. }The identification of $(\gamma,\rho_{b})$ can be similarly established based on the following inequality: \begin{align} & P(q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0)\nonumber\\ \geq & P(\{q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0\}\cap\{I_{im(0,1)}^{+} (\theta)+I_{im(0,1)}^{-}(\theta)=1\})\nonumber\\ \geq & P(q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0|I_{im(0,1)}^{+}(\theta )=I_{im(1,0)}^{-}(\vartheta)=1)P(I_{im(0,1)}^{+}(\theta)=I_{im(1,0)} ^{-}(\vartheta)=1)\nonumber\\ & +P(q_{(1,0)}(Z_{i},Z_{m},\vartheta)>0|I_{im(0,1)}^{-}(\theta)=I_{im(1,0)} ^{+}(\vartheta)=1)P(I_{im(0,1)}^{-}(\theta)=I_{im(1,0)}^{+}(\vartheta )=1), \end{align} where \begin{align*} & P(I_{im(0,1)}^{+}(\theta)=I_{im(1,0)}^{-}(\vartheta)=1)\\ = & P(-X_{im1}^{(1)}\leq u_{im1}(\vartheta)\wedge u_{im1}(\theta ),-X_{im2}^{(1)}\geq u_{im2}(\vartheta)\vee u_{im2}(\theta),u_{imb} (\vartheta)\leq-W_{im}^{(1)}\leq u_{imb}(\theta)), \end{align*} and \begin{align*} & P(I_{im(0,1)}^{-}(\theta)=I_{im(1,0)}^{+}(\vartheta)=1)\\ = & P(-X_{im1}^{(1)}\geq u_{im1}(\vartheta)\vee u_{im1}(\theta),-X_{im2} ^{(1)}\leq u_{im2}(\vartheta)\wedge u_{im2}(\theta),u_{imb}(\theta)\leq -W_{im}^{(1)}\leq u_{imb}(\vartheta)). \end{align*} Finally, put all these results together to conclude that all elements in $\theta$ are identified. \textbf{On ((ref)).} To complete the proof, we need to justify the claim in ((ref)) as well as similar claims for ((ref)) and ((ref)). Since these proofs follow similar arguments, we only present the proof for $P(q_{(1,0)} (Z_{i},Z_{m},\vartheta)>0|I_{im(1,1)}^{+}(\theta)=I_{im(1,0)}^{-} (\vartheta)=1)>0$ and omit the others for the sake of brevity. Note that by definition, when $I_{im(1,0)}^{-}(\vartheta)=1$, $q_{(1,0)}(Z_{i} ,Z_{m},\vartheta)>0$ if and only if $\Delta p_{(1,0)}(Z_{i},Z_{m})>0$. Therefore, we show $P(\Delta p_{(1,0)}(Z_{i},Z_{m})>0|I_{im(1,1)}^{+} (\theta)=I_{im(1,0)}^{-}(\vartheta)=1)>0$ in the following, and the proof is done. Under Assumption CL1(i), $Z_{im}$ has distribution symmetric at 0, and hence we can pick $X_{im2}^{(1)}=W_{im}^{(1)}=0$, $\tilde{X}_{im2}=0$, $\tilde {W}_{im}=0$, and $S_{im}=0$ so that $\{-X_{im2}^{(1)}\leq u_{im2} (\vartheta)\wedge u_{im2}(\theta),-W_{im}^{(1)}\leq u_{imb}(\vartheta)\wedge u_{imb}(\theta)\}$ holds with equalities for any $\vartheta$. Note that here we use the \textquotedblleft matching\textquotedblright\ insight for developing the MRC procedure, that is, we match the utility indexes associated with the second stand-alone alternative and the bundle effect. Meanwhile, under Assumptions CL2(i) and CL2(ii), we can still have $\{u_{im1} (\vartheta)<-X_{im1}^{(1)}<u_{im1}(\theta)\}$ occur with positive probability conditional on $E\equiv\{X_{im2}^{(1)}=W_{im}^{(1)}=0,\tilde{X}_{im2} =0,\tilde{W}_{im}=0,S_{im}=0\}$. In this scenario, inequality ((ref)) implies that $\Delta p_{(1,0)}(Z_{i},Z_{m})>0,$ and say $\Delta p_{(1,0)}(Z_{i},Z_{m})=\delta_{1}>0$ conditional on $E$. Due to the continuity, there exists a small $\delta_{2}>0$ such that conditional on \[ E^{\delta_{2}}=\{ \vert X_{im2}^{(1)}\vert +\vert W_{im}^{(1)}\vert +\Vert \tilde{X}_{im2}\Vert +\Vert \tilde{W}_{im}\Vert +\Vert S_{im}\Vert \leq\delta _{2}\} , \] a small neighborhood of $E$, \[ \Delta p_{(1,0)}(Z_{i},Z_{m})\geq\delta_{1}/2>0. \] Again, due to the continuity, $P\left( E^{\delta_{2}}\right) >0$. Since what we have discussed so far is only one possible scenario for $\Delta p_{(1,0)} (Z_{i},Z_{m})>0$ to occur, we must have $P(q_{(1,0)}(Z_{i},Z_{m} ,\vartheta)>0|I_{im(1,1)}^{+}(\theta)=I_{im(1,0)}^{-}(\vartheta)=1)>P\left( E^{\delta_{2}}\right) >0$. This completes the proof. \end{proof} \subsection{Proof of of Theorem (ref)} \begin{proof}[Proof of Theorem (ref)] Let \[ \hat{Q}_{N(1,0)}^{D}(\vartheta):=\frac{2}{N(N-1)}\sum_{i=1}^{N-1}\sum_{m>i}\hat{q}_{im(1,0)}^{D}(\vartheta), \] where $\hat{q}_{im(1,0)}^{D}(\vartheta)$ is defined in ((ref)). We prove the consistency of $\hat{\theta}$ by verifying the four sufficient conditions in Theorem 2.1 of NeweyMcFadden1994: (S1) $\Theta$ is a compact set, (S2) $\sup_{\vartheta\in\Theta}\vert\hat{Q}_{N(1,0)}^{D}(\vartheta)-Q_{(1,0)}^{D}(\vartheta)\vert=o_{p}(1)$, (S3) $Q_{(1,0)}^{D}(\vartheta)$ is continuous in $\vartheta$, and (S4) $Q_{(1,0)}^{D}(\vartheta)$ is uniquely minimized at $\theta$. Condition (S1) is trivially satisfied under Assumption CL3. The identification condition (S4) is an immediate result of Theorem (ref) and the discussion right after it. The continuity condition (S3) is also satisfied since $Q_{(1,0)}^{D}(\vartheta)$, by definition, can be expressed as the sum of terms of the following type: \begin{align*} \int_{\Omega_{\vartheta}}(\vert1-\Delta p_{(1,0)}(z_{i},z_{m})\vert-1)f_{Z}(z_{i})f_{Z}(z_{m})dz_{i}dz_{m}, \end{align*} where $\Omega_{\vartheta}:=\{x_{im1}'b+s_{im}'\varrho_{1}\geq0,x_{im2}'b+s_{im}'\varrho_{2}\leq0,w_{im}'r+s_{im}'\varrho_{b}\leq0\}$. Then it is easy to see that Assumptions CL1 and CL2 imply the continuity of $Q_{(1,0)}^{D}(\vartheta)$. The remaining task is to show the uniform convergence condition (S2). To this end, we prove $\sup_{\vartheta\in\Theta}\vert\hat{Q}_{N(1,0)}^{D}(\vartheta)-Q_{N(1,0)}^{D}(\vartheta)\vert=o_{p}(1)$ and $\sup_{\vartheta\in\Theta}\vert Q_{N(1,0)}^{D}(\vartheta)-Q_{(1,0)}^{D}(\vartheta)\vert=o_{p}(1)$, where $Q_{N(1,0)}^{D}(\vartheta):=\frac{2}{N(N-1)}\sum_{i=1}^{N-1}\sum_{m>i}q_{(1,0)}^{D}(Z_{i},Z_{m},\vartheta)$. Firstly, we have by definition \begin{align*} \sup_{\vartheta\in\Theta}\vert\hat{Q}_{N(1,0)}^{D}(\vartheta)-Q_{N(1,0)}^{D}(\vartheta)\vert & \leq\frac{2}{N(N-1)}\sum_{i=1}^{N-1}\sum_{m>i}\sup_{\vartheta\in\Theta}\vert\hat{q}_{im(1,0)}^{D}(\vartheta)-q_{(1,0)}^{D}(Z_{i},Z_{m},\vartheta)\vert\\ & \leq\frac{4}{N(N-1)}\sum_{i=1}^{N-1}\sum_{m>i}\vert\Delta\hat{p}_{(1,0)}(Z_{i},Z_{m})-\Delta p_{(1,0)}(Z_{i},Z_{m})\vert\\ & \leq\sup_{(z_{i},z_{m})\in\mathcal{Z}^{2}}4\vert\Delta\hat{p}_{(1,0)}(z_{i},z_{m})-\Delta p_{(1,0)}(z_{i},z_{m})\vert=o_{p}(1), \end{align*} where the last equality follows by Assumption CL5. Secondly, since $\vert\Delta p_{(1,0)}(Z_{i},Z_{m})\vert\leq1$, we can write \begin{align} &\sup_{\vartheta\in\Theta}\vert Q_{N(1,0)}^{D}(\vartheta)-Q_{(1,0)}^{D}(\vartheta)\vert \nonumber \\ \leq &\sup_{\vartheta\in\Theta}\vert\frac{2}{N(N-1)}\sum_{i=1}^{N-1}\sum_{m>i}q_{(1,0)}^{D}(Z_{i},Z_{m},\vartheta)-\mathbb{E}[q_{(1,0)}^{D}(Z_{i},Z_{m},\vartheta)]\vert\nonumber \\ \leq &\sup_{\bar{\vartheta}\in\bar{\Theta}}\vert\frac{2}{N(N-1)}\sum_{i=1}^{N-1}\sum_{m>i}\psi_{(1,0)}^{D}(Z_{i},Z_{m},\bar{\vartheta})-\mathbb{E}[\psi_{(1,0)}^{D}(Z_{i},Z_{m},\bar{\vartheta})]\vert, \end{align} where $\bar{\Theta}:=\{\bar{\vartheta}=(\vartheta,a)|\vartheta\in\Theta,a\in[-1,1]\}$ and \begin{align*} \psi_{(1,0)}^{D}(Z_{i},Z_{m},\bar{\vartheta}):= & [\vert I_{im(1,0)}^{+}(\vartheta)-a\vert+\vert I_{im(1,0)}^{-}(\vartheta)+a\vert]\times[I_{im(1,0)}^{+}(\vartheta)+I_{im(1,0)}^{-}(\vartheta)]\\ & +[1-(I_{im(1,0)}^{+}(\vartheta)+I_{im(1,0)}^{-}(\vartheta))]. \end{align*} Let $\mathcal{G}=\{\psi_{(1,0)}^{D}(\cdot,\cdot,\bar{\vartheta})|\bar{\vartheta}=(\vartheta,a),\vartheta\in\Theta,a\in[-1,1]\}$ define the class of bounded functions $\psi_{(1,0)}^{D}(\cdot,\cdot,\bar{\vartheta})$ indexed by $\bar{\vartheta}$. Using similar arguments in the proof of Lemma (ref), we can show that $\mathcal{G}$ is a Euclidean class\footnote{Alternatively, one can also apply results in Section 2.6 of vdVaartWellner1996 to show that $\mathcal{G}$ is a VC-class.} of functions. Hence, we have \begin{equation} \sup_{\bar{\vartheta}\in\bar{\Theta}}\vert\frac{2}{N(N-1)}\sum_{i=1}^{N-1}\sum_{m>i}\psi_{(1,0)}^{D}(Z_{i},Z_{m},\bar{\vartheta})-\mathbb{E}[\psi_{(1,0)}^{D}(Z_{i},Z_{m},\bar{\vartheta})]\vert=o_{p}(1). \end{equation} Plugging ((ref)) into ((ref)) gives $\sup_{\vartheta\in\Theta}\vert Q_{N(1,0)}^{D}(\vartheta)-Q_{(1,0)}^{D}(\vartheta)\vert=o_{p}(1)$. Then (S2) follows immediately from the triangle inequality, which completes the proof. \end{proof} \section{Proofs for the Panel Data Model} \subsection{Proof of Theorem (ref)} In this section, we provide the proof of Theorem (ref), applying the asymptotic theory developed in SeoOtsu2018. Before the main proof, we first present two supporting lemmas, Lemmas (ref) and (ref), whose proofs are relegated to Appendix (ref). The outline of the proof process is as follows. Lemma (ref) verifies the technical conditions in SeoOtsu2018. Lemma (ref) obtains technical terms for the final asymptotics. Then we apply the results in SeoOtsu2018 and get the asymptotics of our estimators in the proof of Theorem (ref). \begin{lemma} Suppose Assumptions P1--P9 hold. Then $\phi_{Ni}\left( b\right) $ and $\varphi_{Ni}\left( r\right) $ satisfy Assumption M in SeoOtsu2018. \end{lemma} \begin{lemma} Suppose Assumptions P1--P9 hold. Then \[ \lim_{N\rightarrow\infty}( Nh_{N}^{k_{1}+k_{2}}) ^{2/3} \mathbb{E}[ \phi_{Ni}( \beta+\rho\left( Nh_{N}^{k_{1}+k_{2} }) ^{-1/3}\right)] =\frac{1}{2}\rho^{\prime}\mathbb{V}\rho, \] \[ \lim_{N\rightarrow\infty}( N\sigma_{N}^{2k_{1}}) ^{2/3} \mathbb{E}[ \varphi_{Ni}( \gamma+\delta( N\sigma_{N} ^{2k_{1}}) ^{-1/3}) ] =\frac{1}{2}\delta^{\prime }\mathbb{W}\delta, \] \[ \lim_{N\rightarrow\infty} ( Nh_{N}^{k_{1}+k_{2}} ) ^{1/3} \mathbb{E} [ h_{N}^{k_{1}+k_{2}}\phi_{Ni} ( \beta+\rho_{1} ( Nh_{N}^{k_{1}+k_{2}} ) ^{-1/3} ) \phi_{Ni} ( \beta+\rho_{2} ( Nh_{N}^{k_{1}+k_{2}} ) ^{-1/3} ) ] =\mathbb{H}_{1}\left( \rho_{1},\rho_{2}\right) , \] and \[ \lim_{N\rightarrow\infty} ( N\sigma_{N}^{2k_{1}} ) ^{1/3} \mathbb{E} [ \sigma_{N}^{2k_{1}}\varphi_{Ni} ( \gamma+\delta _{1} ( N\sigma_{N}^{2k_{1}} ) ^{-1/3} ) \varphi_{Ni} ( \gamma+\delta_{2} ( N\sigma_{N}^{2k_{1}} ) ^{-1/3} ) ] =\mathbb{H}_{2}\left( \delta_{1},\delta_{2}\right) , \] where $\rho,$ $\rho_{1},$ and $\rho_{2}$ are $k_{1}\times1$ vectors, $\delta,$ $\delta_{1},$ and $\delta_{2}$ are $k_{2}\times1$ vectors, $\mathbb{V}$ is a $k_{1}\times$ $k_{1}$ matrix defined as \begin{align} \mathbb{V}= & -\sum_{t>s}\sum_{d\in\mathcal{D}}\left\{ \int1\left[ x^{\prime}\beta=0\right] \left( \frac{\partial\kappa_{dts}^{\left( 1\right) }\left( x\right) }{\partial x}^{\prime}\beta\right) f_{X_{1ts}|\left\{ X_{2ts}=0,W_{ts}=0\right\} }\left( x\right) xx^{\prime }\text{\emph{d}}\sigma_{0}^{\left( 1\right) }\right. \\ & \left. +\int1\left[ x^{\prime}\beta=0\right] \left( \frac {\partial\kappa_{dts}^{\left( 2\right) }\left( x\right) }{\partial x}^{\prime}\beta\right) f_{X_{2ts}|\left\{ X_{1ts}=0,W_{ts}=0\right\} }\left( x\right) xx^{\prime}\text{\emph{d}}\sigma_{0}^{\left( 2\right) }\right\} ,\nonumber \end{align} $\mathbb{W}$ is a $k_{2}\times$ $k_{2}$ matrix defined as \begin{equation} \mathbb{W}=-\sum_{t>s}\int1\left[ w^{\prime}\gamma=0\right] \left( \frac{\partial\kappa_{\left( 1,1\right) ts}^{\left( 3\right) }\left( w\right) }{\partial w}^{\prime}\gamma\right) f_{W_{ts}|\left\{ X_{1ts}=0,X_{2ts}=0\right\} }\left( w\right) ww^{\prime}\text{\emph{d} }\sigma_{0}^{\left( 3\right) }, \end{equation} and $\mathbb{H}_{1}$ and $\mathbb{H}_{2}$ are written respectively as \begin{align} & \mathbb{H}_{1}\left( \rho_{1},\rho_{2}\right) \\ = & \frac{1}{2}\sum_{t>s}\sum_{d\in\mathcal{D}}\left\{ \int\kappa _{dts}^{\left( 4\right) }\left( \bar{x}\right) \left[ \left\vert \bar {x}^{\prime}\rho_{1}\right\vert +\left\vert \bar{x}^{\prime}\rho _{2}\right\vert -\left\vert \bar{x}^{\prime}\left( \rho_{1}-\rho_{2}\right) \right\vert \right] f_{X_{1ts}|\left\{ X_{2ts}=0,W_{ts}=0\right\} }\left( 0,\bar{x}\right) \text{\emph{d}}\bar{x}f_{X_{2ts},W_{ts}}\left( 0,0\right) \right. \nonumber\\ & \left. +\int\kappa_{dts}^{\left( 5\right) }\left( \bar{x}\right) \left[ \left\vert \bar{x}^{\prime}\rho_{1}\right\vert +\left\vert \bar {x}^{\prime}\rho_{2}\right\vert -\left\vert \bar{x}^{\prime}\left( \rho _{1}-\rho_{2}\right) \right\vert \right] f_{X_{2ts}|\left\{ X_{1ts} =0,W_{ts}=0\right\} }\left( 0,\bar{x}\right) \text{\emph{d}}\bar {x}f_{X_{1ts},W_{ts}}\left( 0,0\right) \right\} \bar{K}_{2}^{k_{1}+k_{2} },\nonumber \end{align} and \begin{align} \mathbb{H}_{2}\left( \delta_{1},\delta_{2}\right) = & \frac{1}{2}\sum _{t>s}\int\kappa_{\left( 1,1\right) ts}^{\left( 6\right) }\left( \bar {w}\right) \left[ \left\vert \bar{w}^{\prime}\delta_{1}\right\vert +\left\vert \bar{w}^{\prime}\delta_{2}\right\vert -\left\vert \bar{w}^{\prime }\left( \delta_{1}-\delta_{2}\right) \right\vert \right] \\ & \cdot f_{W_{ts}|\left\{ X_{1ts}=0,X_{2ts}=0\right\} }\left( 0,\bar {w}\right) \text{\emph{d}}\bar{w}f_{X_{1ts},X_{2ts}}\left( 0,0\right) \bar{K}_{2}^{2k_{1}}.\nonumber \end{align} Technical terms that appear in $\mathbb{V}$, $\mathbb{W}$, $\mathbb{H}_{1}$, $\mathbb{H}_{2}$ are defined as follows. $\sigma_{0}^{\left( 1\right) },$ $\sigma_{0}^{\left( 2\right) },$ and $\sigma_{0}^{\left( 3\right) }$ are the surface measures of $\left\{ X_{1ts}\beta=0\right\} ,$ $\left\{ X_{2ts}\beta=0\right\} $, and $\left\{ W_{ts}\gamma=0\right\} ,$ respectively. $\bar{K}_{2}\equiv\int K\left( u\right) ^{2}$d$u.$ We decompose $X_{1ts},X_{2ts}\ $and $W_{ts}$ into $X_{jts}=a\beta+\bar{X} _{jts},j=1,2,$ and $W_{ts}=a\gamma+\bar{W}_{ts},$ where $\bar{X}_{jts},j=1,2,$ are orthogonal to $\beta,$ and $\bar{W}_{ts}$ is orthogonal to $\gamma$. The density for $X_{1ts}$ is written as $f_{X_{1ts}}\left( a,\bar{X} _{1ts}\right) $ under this decomposition. Densities for $X_{2ts}\ $and $W_{ts}$ are written similarly. Further, \[ \kappa_{dts}^{\left( 1\right) }\left( x\right) \equiv\mathbb{E} [Y_{idst}\left( -1\right) ^{d_{1}}|X_{i1ts}=x,X_{i2ts}=0,W_{its}=0], \] \[ \kappa_{dts}^{\left( 2\right) }\left( x\right) \equiv\mathbb{E} [Y_{idst}\left( -1\right) ^{d_{2}}|X_{i2ts}=x,X_{i1ts}=0,W_{its}=0], \] \[ \kappa_{\left( 1,1\right) ts}^{\left( 3\right) }\left( w\right) \equiv\mathbb{E}\left[ Y_{i(1,1)ts}|W_{its}=w,X_{i1ts}=0,X_{i2ts}=0\right] , \] \[ \kappa_{dts}^{\left( 4\right) }\left( x\right) \equiv\mathbb{E}\left[ \left\vert Y_{idst}\right\vert \text{ }|X_{i1ts}=x,X_{i2ts}=0,W_{its} =0\right] , \] \[ \kappa_{dts}^{\left( 5\right) }\left( x\right) \equiv\mathbb{E}\left[ \left\vert Y_{idst}\right\vert \text{ }|X_{i2ts}=x,X_{i1ts}=0,W_{its} =0\right] , \] and \[ \kappa_{\left( 1,1\right) ts}^{\left( 6\right) }\left( w\right) \equiv\mathbb{E}\left[ \left\vert Y_{i\left( 1,1\right) ts}\right\vert \text{ }|W_{its}=w,X_{i1ts}=0,X_{i2ts}=0\right] . \] \end{lemma} \begin{proof} [Proof of Theorem (ref)]Lemma (ref) verifies the technical conditions for applying the results in SeoOtsu2018, and shows that $\phi_{Ni}\left( b\right) $ and $\varphi_{Ni}\left( r\right) $ are \textit{manageable} in the sense of KimPollard1990. By Assumption P9, Lemma 1 in SeoOtsu2018 and its subsequent analysis, we have \[ \hat{\beta}-\beta=O_{p} ( ( Nh_{N}^{k_{1}+k_{2}} ) ^{-1/3} ) \text{ and }\hat{\gamma}-\gamma=O_{p} ( ( N\sigma_{N}^{2k_{1}} ) ^{-1/3} ) . \] Notice that $\hat{\beta}$ can be equivalently obtained from \[ \arg\max_{b\in\mathcal{B}} ( Nh_{N}^{k_{1}+k_{2}}) ^{2/3}\cdot N^{-1}\sum_{i=1}^{N}\phi_{Ni} ( \beta+ ( Nh_{N}^{k_{1}+k_{2} } ) ^{-1/3} \{ ( Nh_{N}^{k_{1}+k_{2}} ) ^{1/3} ( b-\beta ) \} ) . \] We get the asymptotics of $\hat{\beta}$ if we can get the asymptotics of \[ \arg\max_{\rho\in\mathbb{R}^{k_{1}}} ( Nh_{N}^{k_{1}+k_{2}} ) ^{2/3}\cdot N^{-1}\sum_{i=1}^{N}\phi_{Ni} ( \beta+ ( Nh_{N} ^{k_{1}+k_{2}} ) ^{-1/3}\rho) . \] Since $\phi_{Ni}\left( b\right) $ is \textit{manageable} in the sense of KimPollard1990, by Theorem 1 in SeoOtsu2018 and Lemma (ref), we have the uniform convergence of the above stochastic process and \[ ( Nh_{N}^{k_{1}+k_{2}} ) ^{2/3}\cdot N^{-1}\sum_{i=1}^{N}\phi _{Ni} ( \beta+ ( Nh_{N}^{k_{1}+k_{2}}) ^{-1/3}\rho ) \rightsquigarrow\mathcal{Z}_{1}\left( \rho\right) , \] where $\mathcal{Z}_{1}\left( \rho\right) $ is a Gaussian process with continuous sample path, expected value $\frac{1}{2}\rho^{\prime}\mathbb{V} \rho$ and covariance kernel $\mathbb{H}_{1}\left( \rho_{1},\rho_{2}\right) .$ Apply the CMT to obtain \[ ( Nh_{N}^{k_{1}+k_{2}}) ^{1/3}( \hat{\beta}-\beta) \overset{d}{\rightarrow }\arg\max_{\rho}\mathcal{Z}_{1}\left( \rho\right) . \] Apply similar arguments to $\hat{\gamma}$. By Theorem 1 in SeoOtsu2018 and Lemma (ref), we get \[ (N\sigma_{N}^{2k_{1}})^{1/3}\left( \hat{\gamma}-\gamma\right) \overset{d}{\rightarrow}\arg\max_{\delta\in\mathbb{R}^{k_{2}}}\mathcal{Z} _{2}\left( \delta\right) , \] where $\mathcal{Z}_{2}\left( \delta\right) $ is a Gaussian process with continuous sample path, expected value $\frac{1}{2}\delta^{\prime} \mathbb{W}\delta$ and covariance kernel $\mathbb{H}_{2}\left( \delta _{1},\delta_{2}\right) .$ \end{proof} \subsection{Proof of Theorem (ref)} \begin{proof} [Proof of Theorem (ref)] We show that the numerical bootstrap works for $\beta $. The discussion for $\gamma $ is omitted due to the similarity. The key is to show the results in HongLi2020 hold by modifying condition (vi) in Theorem 4.1 (using their notations), for example, for $ \beta $, that \begin{equation*} \Sigma _{1/2}\left( \rho _{1},\rho _{2}\right) =\lim_{N\rightarrow \infty }(Nh_{N}^{k_{1}+k_{2}})^{1/3}\mathbb{E}[h_{N}^{k_{1}+k_{2}}\phi _{Ni}(\beta +\rho _{1}(Nh_{N}^{k_{1}+k_{2}})^{-1/3})\phi _{Ni}(\beta +\rho _{2}(Nh_{N}^{k_{1}+k_{2}})^{-1/3})] \end{equation*} exists for all $\rho _{1},\rho _{2}\in \mathbb{R}^{k_{1}}$. This is indeed the case as shown by our Lemma (ref) in Appendix (ref). We now provide the details of the proof. By some standard calculation as SeoOtsu2018, we can show that the convergence rate of $\hat{\beta}^{\ast }$ is $(\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}})^{-1/3}.$ Thus, if we manage to show the limit of \begin{align} & (\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}})^{2/3}\cdot N^{-1}\sum_{i=1}^{N}\phi _{Ni}(\beta +\rho (\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}})^{-1/3})+(\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}})^{2/3}\cdot (N\varepsilon _{N1})^{1/2} \\ & \cdot \lbrack N^{-1}\sum_{i=1}^{N}\phi _{Ni}^{\ast }(\beta +\rho (\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}})^{-1/3})-N^{-1}\sum_{i=1}^{N}\phi _{Ni}(\beta +\rho (\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}})^{-1/3})], \notag \end{align} then the limiting distribution of $\hat{\beta}^{\ast }$ can be established. For the first term in equation ((ref)), \begin{align} & ( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{2/3}\cdot N^{-1}\sum_{i=1}^{N}\phi_{Ni}( \beta+\rho( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{-1/3}) \\ = &( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{2/3}\mathbb{E}[ \phi_{Ni}( \beta+\rho( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{-1/3}) ] \notag \\ & +( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{2/3}\cdot N^{-1}\sum_{i=1}^{N}[ \phi_{Ni}( \beta+\rho( \varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{-1/3}) -\mathbb{E}[ \phi_{Ni}( \beta+\rho( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{-1/3}) ] ] \notag \\ = &\frac{1}{2}\rho^{\prime}\mathbb{V}\rho+O_{p}( ( \varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{2/3}\cdot N^{-1/2}\cdot h_{N}^{-( k_{1}+k_{2}) /2}( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{-1/6}) =\frac{1 }{2}\rho^{\prime}\mathbb{V}\rho+o_{p}( 1) , \notag \end{align} where the second equality follows by Lemma (ref), the assumption on $ \varepsilon_{N1}$ such that the bias term $h_{N}^{2}$ is a small order term compared to $( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{2/3},$ and Markov inequality, and the last equality holds by $N\varepsilon_{N1}\rightarrow \infty.$ The second term in equation ((ref)) is a sample average of mean 0 series under $P^{\ast}$ multiplied by $( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{2/3}( N\varepsilon_{N1}) ^{1/2}.$ The covariance kernel of the second term is \begin{align} & ( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{4/3}\cdot( N\varepsilon_{N1}) \cdot N^{-1}\cdot\mathbb{E}^{\ast}[ \phi _{Ni}^{\ast}( \beta+\rho_{1}( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{-1/3}) \phi_{Ni}^{\ast}( \beta+\rho _{2}( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{-1/3}) ] \notag \\ = &( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{1/3}\mathbb{E}^{\ast}[ h_{N}^{k_{1}+k_{2}}\phi_{Ni}^{\ast}( \beta +\rho_{1}( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{-1/3}) \phi_{Ni}^{\ast}( \beta+\rho_{2}( \varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{-1/3}) ] \notag \\ = &( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{1/3}\mathbb{E}[ h_{N}^{k_{1}+k_{2}}\phi_{Ni}( \beta+\rho_{1}( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{-1/3}) \phi _{Ni}( \beta+\rho_{2}( \varepsilon_{N1}^{-1}h_{N}^{k_{1}+k_{2}}) ^{-1/3}) ] +o_{p}( 1) \notag \\ = &\mathbb{H}_{1}( \rho_{1},\rho_{2}) +o_{p}( 1) , \end{align} by the i.i.d. sampling and Lemma (ref). Equations ((ref)) and ((ref)) imply that the term in equation ((ref)) converges to $\mathcal{Z}_{1}^{\ast }\left( \rho \right) $ with mean $\frac{1}{2}\rho ^{\prime }\mathbb{V}\rho $ and covariance kernel $\mathbb{H}_{1}\left( \rho _{1},\rho _{2}\right) $. Thus, $\mathcal{Z}_{1}^{\ast }\left( \rho \right) $ is an independent copy of $\mathcal{Z}_{1}\left( \rho \right) $ and \begin{equation*} (\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}})^{1/3}(\hat{\beta}^{\ast }-\beta ) \overset{d}{\rightarrow }\arg \max_{\rho \in \mathbb{R}^{k_{1}}}\mathcal{Z} _{1}^{\ast }\left( \rho \right) . \end{equation*} Finally, note, by $N\varepsilon _{N1}\rightarrow \infty $ and \begin{align*} (\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}})^{1/3}(\hat{\beta}^{\ast }-\hat{ \beta})& =(\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}})^{1/3}(\hat{\beta} ^{\ast }-\beta )+(\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}})^{1/3}(\hat{\beta }-\beta ) \\ & =(\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}})^{1/3}(\hat{\beta}^{\ast }-\beta )+O_{p}((\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}})^{1/3}\cdot (Nh_{N}^{k_{1}+k_{2}})^{-1/3}) \\ & =(\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}})^{1/3}(\hat{\beta}^{\ast }-\beta )+o_{p}(1), \end{align*} we have \begin{equation*} (\varepsilon _{N1}^{-1}h_{N}^{k_{1}+k_{2}})^{1/3}(\hat{\beta}^{\ast }-\hat{ \beta})\overset{d}{\rightarrow }\arg \max_{\rho \in \mathbb{R}^{k_{1}}} \mathcal{Z}_{1}^{\ast }\left( \rho \right) . \end{equation*} \end{proof} \subsection{Proof of Theorem (ref)} \begin{proof} [Proof of Theorem (ref)] Recall that under Assumptions P1--P9, we have shown that our estimator satisfies the technical conditions required by SeoOtsu2018 in Lemma (ref). Then applying Lemma 1 in SeoOtsu2018 to $N^{-1}\mathcal{L}_{N,\gamma }^{P,K}(r),$ we get \begin{equation} N^{-1}\mathcal{L}_{N,\gamma }^{P,K}(r)-\mathcal{\bar{L}}^{P}\left( r\right) =N^{-1}\mathcal{L}_{N,\gamma }^{P,K}(\gamma )-\mathcal{\bar{L}}^{P}\left( \gamma \right) +\varepsilon \left\Vert r-\gamma \right\Vert ^{2}+O_{p}( \sigma _{N}^{2}) +O_{p}( ( N\sigma _{N}^{2k_{1}}) ^{-2/3}) \end{equation} for any small $\varepsilon >0,$ where use some of the results in Lemma (ref). Some standard calculations as in the proof of Lemma (ref) yield \begin{equation} \mathcal{\bar{L}}^{P}\left( r\right) -\mathcal{\bar{L}}^{P}\left( \gamma \right) =\frac{1}{2}\left( r-\gamma \right) ^{\prime }\mathbb{W}\left( r-\gamma \right) +o_{p}\left( \left\Vert r-\gamma \right\Vert ^{2}\right) \end{equation} for any $r$ in a small neighborhood of $\gamma .$ Recall that we assume $( N\sigma _{N}^{2k_{1}})^{2/3}\sigma _{N}^{2}\rightarrow 0$ in Assumption P9 and that $\mathbb{W}$ is finite. Combining these two conditions with ((ref)) and ((ref)) gives \begin{equation*} N^{-1}\mathcal{L}_{N,\gamma }^{P,K}(r)=N^{-1}\mathcal{L}_{N,\gamma }^{P,K}(\gamma )+O_{p}\left( \left\Vert r-\gamma \right\Vert ^{2}\right) +O_{p}( \left( N\sigma _{N}^{2k_{1}}\right) ^{-2/3}) , \end{equation*} which further implies \begin{equation*} N^{-1}\mathcal{L}_{N,\gamma }^{P,K}(\hat{\gamma})-N^{-1}\mathcal{L} _{N,\gamma }^{P,K}(\gamma )=O_{p}( \left( N\sigma _{N}^{2k_{1}}\right) ^{-2/3}) \end{equation*} since $\hat{\gamma}-\gamma =O_{p}( \left( N\sigma _{N}^{2k_{1}}) ^{-1/3}\right)$ as shown in Theorem (ref). Therefore, we have \begin{equation} \sqrt{N\sigma _{N}^{2k_{1}}}\left( N^{-1}\mathcal{L}_{N,\gamma }^{P,K}(\hat{ \gamma})-N^{-1}\mathcal{L}_{N,\gamma }^{P,K}(\gamma )\right) =o_{p}\left( 1\right) . \end{equation} Additionally, by applying the Lindeberg-Feller CLT, we have \begin{equation*} \sqrt{N\sigma _{N}^{2k_{1}}}\left( N^{-1}\mathcal{L}_{N,\gamma }^{P,K}(\gamma )-\mathcal{\bar{L}}^{P}\left( \gamma \right) \right) \overset{ d}{\rightarrow }N\left( 0,\Delta ^{P}_{\gamma}\right) . \end{equation*} Combining this with ((ref)), we arrive at the desired result \begin{equation*} \sqrt{N\sigma _{N}^{2k_{1}}}\left( N^{-1}\mathcal{L}_{N,\gamma }^{P,K}(\hat{ \gamma})-\mathcal{\bar{L}}^{P}\left( \gamma \right) \right) \overset{d}{ \rightarrow }N\left( 0,\Delta ^{P}_{\gamma}\right) . \end{equation*} \end{proof} \section{Technical Details for Theorem (ref)} In this section, we show the validity of the nonparametric bootstrap for the cross-sectional model (Section (ref)). A complete proof of the consistency of the bootstrap procedure (Theorem (ref)) is lengthy and tedious. Considering that this process is relatively standard in the literature, we only present an outline of the proof. To fill in the gaps of the outline, one first needs to define a product probability space for both the original random series and the bootstrap series as in WellnerZhan1996 (an application can be found in AbrevayaHuang2005), define a suitable norm for functions, and then establish some uniform convergence results following Sherman1993, Sherman1994AoS, Sherman1994ET on this probability space with this norm. To apply Sherman's results, the criterion functions need to be \textit{manageable} in the sense of KimPollard1990. This is indeed the case for our estimators. For example, equation ((ref)) is a summation of \[ \mathcal{K}_{\sigma_{N}}( V_{im} ( \hat{\beta} ) ) Y_{im\left( 1,1\right) }\text{sgn}\left( W_{im}^{\prime}r\right) =\mathcal{K}_{\sigma_{N}} ( V_{im} ( \hat{\beta} ) ) Y_{im\left( 1,1\right) }\left( 2\cdot1\left[ W_{im}^{\prime}r>0\right] -1\right) . \] The collection of indicator functions $1\left[ W_{im}^{\prime}r>0\right] $ for $r\in\mathcal{R}$\ is well known to be a Vapnik--Chervonenkis (VC) class. The kernel function $\mathcal{K}$, when is chosen to satisfy some mild conditions (e.g., uniformly bounded with compact support and continuously differentiable), is also \textit{manageable}. The complication of the bandwidth can be handled in the same way as in SeoOtsu2018. Now suppose we have handled the technical details mentioned above. In what follows, we present the proof outline. To ease expositions, we assume $\hat{\beta}$ and $\hat{\gamma}$ are obtained from maximizing$,$ respectively, $\mathbb{L}_{N,\beta}^{K}\left( b\right) $ and $\mathbb{L}_{N,\gamma}^{K} ( r,\hat{\beta} ) $, which are defined as \[ \mathbb{L}_{N,\beta}^{K}\left( b\right) =\sum_{i=1}^{N-1}\sum_{m>i} \mathcal{K}_{h_{N}}\left( X_{im}\right) \chi_{1}\left( Z_{i},Z_{m},b\right) \] and \[ \mathbb{L}_{N,\gamma}^{K} ( r,\hat{\beta} ) =\sum_{i=1}^{N-1} \sum_{m>i}\mathcal{K}_{\sigma_{N}} ( X_{im}^{\prime}\hat{\beta} ) \chi_{2}\left( Z_{i},Z_{m},r\right), \] where $X_{im}$ is a sub-vector of $Z_i-Z_m$. Note that the criterion functions and notations defined here are meant for general U-processes, and thus are different from the ones used in the main text. However, the procedures that we study here share the same structure as those in Section (ref) (i.e., ((ref))--((ref))). Therefore, if we can demonstrate the consistency of the bootstrap here, we can easily apply the same method to our MRC estimators. We do normalization such that $\chi_{1}\left( Z_{m},Z_{i},\beta\right) =0$ and $\chi_{2}\left( Z_{i},Z_{m},\gamma\right) =0$ for all $Z.$ \footnote{Taking equation ((ref)) as an example, this can be done by replacing $\text{sgn}\left( W^{\prime}_{im}r\right) $ with $\text{sgn}\left( W^{\prime}_{im}r\right) -\text{sgn}\left( W^{\prime}_{im}\gamma\right) $ in the criterion function.} Assume that $\chi_{1}\left( Z_{i},Z_{m},b\right) =\chi_{1}\left( Z_{m},Z_{i},b\right) ,$ $\beta$ uniquely maximizes $\mathbb{E}\left[ \chi_{1}\left( Z_{i},Z_{m},b\right) |X_{im}=0\right] ,$ and $\mathcal{K}_{h_{N}}$ is a $p$-th order kernel with $h_{N}$ that satisfies $Nh_{N}^{p}=o\left( 1\right) .$ Similarly, we assume that $\chi_{2}\left( Z_{i},Z_{m},r\right) =\chi_{2}\left( Z_{m},Z_{i},r\right) ,$ $\gamma$ uniquely maximizes $\mathbb{E}\left[ \chi_{2}\left( Z_{i},Z_{m},r\right) |X_{im}^{\prime}\beta=0\right] ,$ and $\mathcal{K}_{\sigma_{N}}$ is a $p$-th order kernel with $\sigma_{N}$ that satisfies $N\sigma_{N}^{p}=o\left( 1\right) .$ We similarly denote $\mathbb{L}_{N,\beta}^{K\ast}\left( b\right) $ and $\mathbb{L}_{N,\gamma}^{K\ast} ( r,\hat{\beta}^{\ast} ) $ as the corresponding criterion functions using the bootstrap series. Before establishing the consistency of the bootstrap, it is helpful to do a recast of how we derive the asymptotics for $( \hat{\beta},\hat{\gamma }) .$ $\mathbb{L}_{N,\beta}^{K}\left( b\right) $ can be decomposed into \begin{align} & 2\left[ N\left( N-1\right) \right] ^{-1}\mathbb{L}_{N,\beta}^{K}\left( b\right) \\ = &\mathbb{E}\left[ \mathcal{K}_{h_{N}}\left( X_{im}\right) \chi _{1}\left( Z_{i},Z_{m},b\right) \right] \nonumber\\ & +\frac{2}{N}\sum_{i=1}^{N}\left\{ \mathbb{E}\left[ \mathcal{K}_{h_{N} }\left( X_{im}\right) \chi_{1}\left( Z_{i},Z_{m},b\right) |Z_{i}\right] -\mathbb{E}\left[ \mathcal{K}_{h_{N}}\left( X_{im}\right) \chi_{1}\left( Z_{i},Z_{m},b\right) \right] \right\} \nonumber\\ & +\frac{1}{N\left( N-1\right) }\sum_{i=1}^{N}\sum_{m=1,m\neq i} ^{N}\left\{ \mathcal{K}_{h_{N}}\left( X_{im}\right) \chi_{1}\left( Z_{i},Z_{m},b\right) -\mathbb{E}\left[ \mathcal{K}_{h_{N}}\left( X_{im}\right) \chi_{1}\left( Z_{i},Z_{m},b\right) |Z_{i}\right] \right. \nonumber\\ & \left. -\mathbb{E}\left[ \mathcal{K}_{h_{N}}\left( X_{im}\right) \chi_{1}\left( Z_{i},Z_{m},b\right) |Z_{m}\right] +\mathbb{E}\left[ \mathcal{K}_{h_{N}}\left( X_{im}\right) \chi_{1}\left( Z_{i},Z_{m} ,b\right) \right] \right\} \nonumber\\ \equiv &\mathcal{T}_{N1}+\mathcal{T}_{N2}+\mathcal{T}_{N3}.\nonumber \end{align} Since $\chi_{1}\left( Z_{i},Z_{m},\beta\right) =0$ and $\beta$ uniquely maximizes $\mathbb{E}\left[ \chi_{1}\left( Z_{i},Z_{m},b\right) |X_{im}=0\right] ,$ we have \begin{align*} \mathcal{T}_{1N} & =\mathbb{E}\left[ \chi_{1}\left( Z_{i},Z_{m},b\right) |X_{im}=0\right] +O_{p}\left( h_{N}^{p}\right) \\ & =\frac{1}{2}\left( b-\beta\right) ^{\prime}\mathbb{V}_{\beta}\left( b-\beta\right) +O_{p}\left( \left\Vert b-\beta\right\Vert ^{3}\right) +O_{p}\left( h_{N}^{p}\right) , \end{align*} where the $O_{p} ( h_{N}^{p} ) $ is the bias term from matching and $\mathbb{V}_{\beta}$ is a negative definite matrix. For the second term, \begin{align} \mathcal{T}_{N2} & =\frac{2}{N}\sum_{i=1}^{N}\left\{ \mathbb{E}\left[ \chi_{1}\left( Z_{i},Z_{m},b\right) |Z_{i},X_{im}=0\right] f_{X_{m}}\left( X_{i}\right) -\mathbb{E}\left[ \chi_{1}\left( Z_{i},Z_{m},b\right) |X_{im}=0\right] \right\} +O_{p}\left( h_{N}^{p}\right) \nonumber\\ & =2\left( b-\beta\right) ^{\prime}\frac{W_{N1}}{\sqrt{N}}+o_{p}\left( \left\Vert b-\beta\right\Vert ^{2}\right) +O_{p}\left( h_{N}^{p}\right) , \end{align} where $W_{N1}$ converges to a normal distribution. For the last term, \[ \mathcal{T}_{N3}=o_{p}\left( N^{-1}\right) , \] by similar arguments in Sherman1993. Put all these results together to obtain \[ 2\left[ N\left( N-1\right) \right] ^{-1}\mathbb{L}_{N,\beta}^{K}\left( b\right) =\frac{1}{2}\left( b-\beta\right) ^{\prime}\mathbb{V}_{\beta }\left( b-\beta\right) +2\left( b-\beta\right) ^{\prime}\frac{W_{N1} }{\sqrt{N}}+o_{p}\left( \left\Vert b-\beta\right\Vert ^{2}\right) +o_{p}\left( N^{-1}\right) . \] $\hat{\beta}-\beta$ can be shown to be $O_{p} ( N^{-1/2} ) $. Using the rate result and invoking the CMT, we have \begin{equation} \sqrt{N} ( \hat{\beta}-\beta ) =-2\mathbb{V}_{\beta}^{-1} W_{N1}+o_{p}\left( 1\right) . \end{equation} Let $\mathbb{E}^{\ast}$ and $P^{\ast}$\ denote the expectation and the density for the bootstrap series, respectively. By definition, $P^{\ast}$ puts $N^{-1}$ probability on each of $\left\{ Z_{i}\right\} _{i=1}^{N}.$ We can do a similar decomposition of $\mathbb{L}_{N,\beta}^{K\ast}\left( b\right) $ as for $\mathbb{L}_{N,\beta}^{K}\left( b\right) :$ \begin{align} & 2\left[ N\left( N-1\right) \right] ^{-1}\mathbb{L}_{N,\beta}^{K\ast }\left( b\right) \nonumber\\ = &\left[ N\left( N-1\right) \right] ^{-1}\sum_{i=1}^{N}\sum_{m=1,m\neq i}^{N}\mathcal{K}_{h_{N}}\left( X_{im}^{\ast}\right) \chi_{1}\left( Z_{i}^{\ast},Z_{m}^{\ast},b\right) \nonumber\\ = &\mathbb{E}^{\ast}\left[ \mathcal{K}_{h_{N}}\left( X_{im}^{\ast}\right) \chi_{1}\left( Z_{i}^{\ast},Z_{m}^{\ast},b\right) \right] \nonumber\\ & +\frac{2}{N}\sum_{i=1}^{N}\left\{ \mathbb{E}^{\ast}\left[ \mathcal{K} _{h_{N}}\left( X_{im}^{\ast}\right) \chi_{1}\left( Z_{i}^{\ast},Z_{m} ^{\ast},b\right) |Z_{i}^{\ast}\right] -\mathbb{E}^{\ast}\left[ \mathcal{K}_{h_{N}}\left( X_{im}^{\ast}\right) \chi_{1}\left( Z_{i}^{\ast },Z_{m}^{\ast},b\right) \right] \right\} \nonumber\\ & +\frac{1}{N\left( N-1\right) }\sum_{i=1}^{N}\sum_{m=1,m\neq i} ^{N}\left\{ \mathcal{K}_{h_{N}}\left( X_{im}^{\ast}\right) \chi_{1}\left( Z_{i}^{\ast},Z_{m}^{\ast},b\right) -\mathbb{E}^{\ast}\left[ \mathcal{K} _{h_{N}}\left( X_{im}^{\ast}\right) \chi_{1}\left( Z_{i}^{\ast},Z_{m} ^{\ast},b\right) |Z_{i}^{\ast}\right] \right. \nonumber\\ & \left. -\mathbb{E}^{\ast}\left[ \mathcal{K}_{h_{N}}\left( X_{im}^{\ast }\right) \chi_{1}\left( Z_{i}^{\ast},Z_{m}^{\ast},b\right) |Z_{m}^{\ast }\right] +\mathbb{E}^{\ast}\left[ \mathcal{K}_{h_{N}}\left( X_{im}^{\ast }\right) \chi_{1}\left( Z_{i}^{\ast},Z_{m}^{\ast},b\right) \right] \right\} \nonumber\\ \equiv &\mathcal{T}_{N1}^{\ast}+\mathcal{T}_{N2}^{\ast}+\mathcal{T} _{N3}^{\ast}. \end{align} For $\mathcal{T}_{N1}^{\ast},$ by the definition of of $\mathbb{E}^{\ast},$ \begin{align} \mathcal{T}_{N1}^{\ast} = &\mathbb{E}^{\ast}\left[ \mathcal{K}_{h_{N} }\left( X_{im}^{\ast}\right) \chi_{1}\left( Z_{i}^{\ast},Z_{m}^{\ast },b\right) \right] \nonumber \\ =&\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{m=1}^{N}\mathcal{K}_{h_{N}}\left( X_{im}\right) \chi_{1}\left( Z_{i},Z_{m},b\right) \nonumber\\ = &\frac{1}{2}\left( b-\beta\right) ^{\prime}\mathbb{V}_{\beta}\left( b-\beta\right) +2\left( b-\beta\right) ^{\prime}\frac{W_{N1}}{\sqrt{N} }+O_{p}\left( \left\Vert b-\beta\right\Vert ^{3}\right) +o_{p}\left( N^{-1}\right) , \end{align} where the last line holds because the second line is the same as $2\left[ N\left( N-1\right) \right] ^{-1}\mathbb{L}_{N,\beta}^{K}\left( b\right) $ plus a small order term. Both $\mathcal{T}_{N2}$ and $\mathcal{T}_{N2}^{\ast}$ are sample averages of mean zero series. Moreover,\textbffor $\left\Vert b-\beta\right\Vert =O ( N^{-1/2} ) ,$ $N\mathcal{T}_{N2}=2\sqrt{N}\left( b-\beta\right) ^{\prime}W_{N1}+o_{p}\left( 1\right) $ and $W_{N1}$ converges to a normal distribution$.$ Then, we can apply the results in gine1990bootstrapping that $NT_{N2}^{\ast}$ approximates the distribution of $NT_{N2}$ for $\left\Vert b-\beta\right\Vert =O ( N^{-1/2} ) .$ That is, for $\left\Vert b-\beta\right\Vert =O ( N^{-1/2} ) ,$ \begin{equation} \mathcal{T}_{N2}^{\ast}=2\left( b-\beta\right) ^{\prime}\frac{W_{N1}^{\ast} }{\sqrt{N}}+o_{p}\left( N^{-1}\right) , \end{equation} where $W_{N1}^{\ast}$ is an independent copy of $W_{N1}.$ For $\mathcal{T}_{N3}^{\ast},$ terms across $i$ and $m$ are uncorrelated with each other under $P^{\ast}$. Therefore, $\left[ \mathbb{E}^{\ast}\left( \mathcal{T}_{N3}^{\ast2}\right) \right] ^{1/2}$ is of the same order as \begin{align*} & N^{-1}\left\{ \mathbb{E}^{\ast}\left[ \left[ \mathcal{K}_{h_{N}}\left( X_{im}^{\ast}\right) \chi_{1}\left( Z_{i}^{\ast},Z_{m}^{\ast},b\right) \right] ^{2}\right] \right\} ^{1/2} =N^{-1}\left\{ \frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{m=1}^{N}\mathcal{K} _{h_{N}}\left( X_{im}\right) ^{2}\chi_{1}\left( Z_{i},Z_{m},b\right) ^{2}\right\}^{1/2}. \end{align*} Note that $\chi_{1}\left( Z_{i},Z_{m},\beta\right) =0$, so the above term is $o_{p}\left( N^{-1}\right) $ for $\left\Vert b-\beta\right\Vert =O\left( N^{-1/2}\right) .$ That means $\left[ \mathbb{E}^{\ast}\left( \mathcal{T}_{N3}^{\ast2}\right) \right] ^{1/2}=o_{p}\left( N^{-1}\right) .$ By Markov inequality, \begin{equation} \mathcal{T}_{N3}^{\ast}=o_{p}\left( N^{-1}\right) . \end{equation} To summarize, equations ((ref))--((ref)) imply that for $\left\Vert b-\beta\right\Vert =O\left( N^{-1/2}\right) ,$ \[ 2\left[ N\left( N-1\right) \right] ^{-1}\mathbb{L}_{N,\beta}^{K\ast }\left( b\right) =\frac{1}{2}\left( b-\beta\right) ^{\prime} \mathbb{V}_{\beta}\left( b-\beta\right) +2\left( b-\beta\right) ^{\prime }\frac{W_{N1}}{\sqrt{N}}+2\left( b-\beta\right) ^{\prime}\frac{W_{N1}^{\ast }}{\sqrt{N}}+o_{p}\left( N^{-1}\right) . \] We can similarly show that $\hat{\beta}^{\ast}-\beta=O_{p}\left( N^{-1/2}\right) .$ By the CMT again, \[ \sqrt{N} ( \hat{\beta}^{\ast}-\beta) =-2\mathbb{V}_{\beta} ^{-1}W_{N1}-2\mathbb{V}_{\beta}^{-1}W_{N1}^{\ast}+o_{p}\left( 1\right) . \] Substitute equation ((ref)) into the equation above to get \begin{equation} \sqrt{N} ( \hat{\beta}^{\ast}-\hat{\beta} ) =-2\mathbb{V}_{\beta }^{-1}W_{N1}^{\ast}+o_{p}\left( 1\right) . \end{equation} Since $W_{N1}^{\ast}$ is an independent copy of $W_{N1},$ $\sqrt{N} ( \hat{\beta}^{\ast}-\hat{\beta} ) $ converges to the same distribution as the one $\sqrt{N} ( \hat{\beta}-\beta ) $ converges to. We turn to the inference for $\hat{\gamma}.$ Again, we decompose $\mathbb{L}_{N,\gamma}^{K} ( r,\hat{\beta} ) $ and analyze each term in the decompositions one by one. We start with \begin{align} & 2\left[ N\left( N-1\right) \right] ^{-1}\mathbb{L}_{N,\gamma} ^{K}\left( r,\hat{\beta}\right) \\ = & \left[ N\left( N-1\right) \right] ^{-1}\sum_{i=1}^{N}\sum_{m=1,m\neq i}^{N}\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime}\hat{\beta}\right) \chi_{2}\left( Z_{i},Z_{m},r\right) \nonumber\\ = &\left[ N\left( N-1\right) \right] ^{-1}\sum_{i=1}^{N}\sum_{m=1,m\neq i}^{N}\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime}\beta\right) \chi _{2}\left( Z_{i},Z_{m},r\right) \nonumber\\ & +\left[ N\left( N-1\right) \right] ^{-1}\sum_{i=1}^{N}\sum_{m=1,m\neq i}^{N}\chi_{2}\left( Z_{i},Z_{m},r\right) \nabla\mathcal{K}_{\sigma_{N} }\left( X_{im}^{\prime}\beta\right) X_{im}^{\prime}\frac{\left( \hat{\beta }-\beta\right) }{\sigma_{N}}\nonumber\\ & +\left[ N\left( N-1\right) \right] ^{-1}\sum_{i=1}^{N}\sum_{m=1,m\neq i}^{N}\chi_{2}\left( Z_{i},Z_{m},r\right) \nabla^{2}\mathcal{K}_{\sigma_{N} }\left( X_{im}^{\prime}\tilde{\beta}\right) \left[ X_{im}^{\prime} \frac{\left( \hat{\beta}-\beta\right) }{\sigma_{N}}\right] ^{2}\nonumber\\ \equiv & \mathcal{T}_{N4}+\mathcal{T}_{N5}+\mathcal{T}_{N6},\nonumber \end{align} where $\tilde{\beta}$ is a vector that lies between $\beta$ and $\hat {\beta}$ and makes the equality hold. Note that $\mathcal{T}_{N4}$ shares the same structure as $2\left[ N\left( N-1\right) \right] ^{-1}\mathbb{L}_{N,\beta}^{K}\left( b\right) ,$ and so \begin{equation} \mathcal{T}_{N4}=\frac{1}{2}\left( r-\gamma\right) ^{\prime}\mathbb{V} _{\gamma}\left( r-\gamma\right) +2\left( r-\gamma\right) ^{\prime} \frac{W_{N2}}{\sqrt{N}}+O_{p}\left( \left\Vert r-\gamma\right\Vert ^{3}\right) +O_{p}\left( \sigma_{N}^{p}\right) +o_{p}\left( N^{-1}\right) , \end{equation} where $\mathbb{V}_{\gamma}$ is a negative definite matrix, and $W_{N2}$ converges to a normal distribution. For term $\mathcal{T}_{N5},$ \[ \mathcal{T}_{N5}=\left\{ \left[ N\left( N-1\right) \right] ^{-1} \sum_{i=1}^{N}\sum_{m=1,m\neq i}^{N}\chi_{2}\left( Z_{i},Z_{m},r\right) \sigma_{N}^{-1}\nabla\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime} \beta\right) X_{im}^{\prime}\right\} \left( \hat{\beta}-\beta\right) . \] Note that \begin{align} & \left[ N\left( N-1\right) \right] ^{-1}\sum_{i=1}^{N}\sum_{m=1,m\neq i}^{N}\chi_{2}\left( Z_{i},Z_{m},r\right) \sigma_{N}^{-1}\nabla \mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime}\beta\right) X_{im}^{\prime }\\ = &\mathbb{E}\left[ \chi_{2}\left( Z_{i},Z_{m},r\right) \sigma_{N} ^{-1}\nabla\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime}\beta\right) X_{im}^{\prime}\right] \nonumber\\ & +\frac{2}{N}\sum_{i=1}^{N}\left\{ \mathbb{E}\left[ \chi_{2}\left( Z_{i},Z_{m},r\right) \sigma_{N}^{-1}\nabla\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime}\beta\right) X_{im}^{\prime}|Z_{i}\right] -\mathbb{E}\left[ \chi_{2}\left( Z_{i},Z_{m},r\right) \sigma_{N}^{-1}\nabla\mathcal{K} _{\sigma_{N}}\left( X_{im}^{\prime}\beta\right) X_{im}^{\prime}\right] \right\} \nonumber\\ & +\frac{1}{N\left( N-1\right) }\sum_{i=1}^{N}\sum_{m=1,m\neq i} ^{N}\left\{ \chi_{2}\left( Z_{i},Z_{m},r\right) \sigma_{N}^{-1} \nabla\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime}\beta\right) X_{im}^{\prime}-\mathbb{E}\left[ \chi_{2}\left( Z_{i},Z_{m},r\right) \sigma_{N}^{-1}\nabla\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime} \beta\right) X_{im}^{\prime}|Z_{i}\right] \right. \nonumber\\ & \left. -\mathbb{E}\left[ \chi_{2}\left( Z_{i},Z_{m},r\right) \sigma _{N}^{-1}\nabla\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime}\beta\right) X_{im}^{\prime}|Z_{m}\right] +\mathbb{E}\left[ \chi_{2}\left( Z_{i} ,Z_{m},r\right) \sigma_{N}^{-1}\nabla\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime}\beta\right) X_{im}^{\prime}\right] \right\} \nonumber \end{align} with the lead term $\mathbb{E} [ \chi_{2}\left( Z_{i},Z_{m},r\right) \sigma_{N}^{-1}\nabla\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime} \beta\right) X_{im}^{\prime} ] $. As $\int\nabla\mathcal{K} _{\sigma_{N}}\left( u\right) \text{d}u=0,$ the lead term cancels one more $\sigma_{N}^{-1}$ when calculating the expectation. Since $\chi_{2}\left( Z_{i},Z_{m},\gamma\right) =0,$ we can write \begin{equation} \mathbb{E}\left[ \chi_{2}\left( Z_{i},Z_{m},r\right) \sigma_{N}^{-1} \nabla\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime}\beta\right) X_{im}\right] =\left( r-\gamma\right) ^{\prime}\Gamma+O ( \left\Vert r-\gamma\right\Vert ^{2} ) +O\left( \sigma_{N}^{p}\right) , \end{equation} for some matrix $\Gamma.$ Collect all these result to obtain \begin{align} \mathcal{T}_{N5} & =\left( r-\gamma\right) ^{\prime}\Gamma\left( \hat{\beta}-\beta\right) +O_{p}\left( \left\Vert r-\gamma\right\Vert ^{2}\left\Vert \hat{\beta}-\beta\right\Vert \right) +O\left( \sigma_{N} ^{p}\right) \\ & =-2\left( r-\gamma\right) ^{\prime}\Gamma\mathbb{V}_{\beta}^{-1} \frac{W_{N1}}{\sqrt{N}}+O_{p}\left( \left\Vert r-\gamma\right\Vert ^{2}\left\Vert \hat{\beta}-\beta\right\Vert \right) +o_{p}\left( \frac{\left\Vert r-\gamma\right\Vert }{\sqrt{N}}\right) +O\left( \sigma _{N}^{p}\right), \nonumber \end{align} where the second line follows by substituting in equation ((ref)). For term $\mathcal{T}_{N6},$ it is not hard to see that \begin{equation} \mathcal{T}_{N6}=O_{p}\left( \left. \left\Vert r-\gamma\right\Vert \left\Vert \hat{\beta}-\beta\right\Vert ^{2}\right/ \sigma_{N}^{2}\right) =o_{p}\left( N^{-1}\right) , \end{equation} for $\Vert r-\gamma\Vert=O ( N^{-1/2} ) $, and $\sigma_{N}$ in Assumption C7. To summarize, equations ((ref)), ((ref)), ((ref)), and ((ref)) imply that for $\left\Vert r-\gamma\right\Vert =O\left( N^{-1/2}\right) ,$ \begin{align*} 2\left[ N\left( N-1\right) \right] ^{-1}\mathbb{L}_{N,\gamma}^{K}\left( r,\hat{\beta}\right) = &\frac{1}{2}\left( r-\gamma\right) ^{\prime }\mathbb{V}_{\gamma}\left( r-\gamma\right) \\ & +2\left( r-\gamma\right) ^{\prime}\frac{W_{N2}}{\sqrt{N}}-2\left( r-\gamma\right) ^{\prime}\Gamma\mathbb{V}_{\beta}^{-1}\frac{W_{N1}}{\sqrt{N} }+o_{p}\left( N^{-1}\right) . \end{align*} Once $\hat{\gamma}-\gamma=O_{p}\left( N^{-1/2}\right)$ is established, applying the CMT gives \begin{equation} \sqrt{N}\left( \hat{\gamma}-\gamma\right) =-2\mathbb{V}_{\gamma}^{-1} W_{N2}+2\mathbb{V}_{\gamma}^{-1}\Gamma\mathbb{V}_{\beta}^{-1}W_{N1} +o_{p}\left( 1\right) . \end{equation} For $\mathbb{L}_{N,\gamma}^{K\ast}\left( r,b\right) ,$ we decompose it similarly: \begin{align} & 2\left[ N\left( N-1\right) \right] ^{-1}\mathbb{L}_{N,\gamma}^{K\ast }\left( r,\hat{\beta}^{\ast}\right) \\ & =\left[ N\left( N-1\right) \right] ^{-1}\sum_{i=1}^{N}\sum_{m=1,m\neq i}^{N}\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\ast\prime}\hat{\beta}^{\ast }\right) \chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast},r\right) \nonumber\\ & =\left[ N\left( N-1\right) \right] ^{-1}\sum_{i=1}^{N}\sum_{m=1,m\neq i}^{N}\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\ast\prime}\hat{\beta}\right) \chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast},r\right) \nonumber\\ & +\left[ N\left( N-1\right) \right] ^{-1}\sum_{i=1}^{N}\sum_{m=1,m\neq i}^{N}\chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast},r\right) \nabla \mathcal{K}_{\sigma_{N}}\left( X_{im}^{\ast\prime}\hat{\beta}\right) \left( \frac{X_{im}^{\ast\prime}\left( \hat{\beta}^{\ast}-\hat{\beta}\right) }{\sigma_{N}}\right) \nonumber\\ & +\left[ N\left( N-1\right) \right] ^{-1}\sum_{i=1}^{N}\sum_{m=1,m\neq i}^{N}\chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast},r\right) \nabla ^{2}\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\ast\prime}\tilde{\beta}^{\ast }\right) \left( \frac{X_{im}^{\ast\prime}\left( \hat{\beta}^{\ast} -\hat{\beta}\right) }{\sigma_{N}}\right) ^{2}\nonumber\\ & \equiv\mathcal{T}_{N4}^{\ast}+\mathcal{T}_{N5}^{\ast}+\mathcal{T} _{N6}^{\ast},\nonumber \end{align} where $\tilde{\beta}^{\ast}$ is some vector that lies between $b$ and $\hat{\beta}$ and makes the equality hold. For $\mathcal{T}_{N4}^{\ast},$ we have \begin{align} \mathcal{T}_{N4}^{\ast} =&\left[ N\left( N-1\right) \right] ^{-1} \sum_{i=1}^{N}\sum_{m=1,m\neq i}^{N}\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\ast\prime}\hat{\beta}\right) \chi_{2}\left( Z_{i}^{\ast} ,Z_{m}^{\ast},r\right) \nonumber\\ = &\mathbb{E}^{\ast}\left[ \mathcal{K}_{\sigma_{N}}\left( X_{im} ^{\ast\prime}\hat{\beta}\right) \chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast },r\right) \right] \nonumber\\ & +\frac{2}{N}\sum_{i=1}^{N}\left\{ \mathbb{E}^{\ast}\left[ \mathcal{K} _{\sigma_{N}}\left( X_{im}^{\ast\prime}\hat{\beta}\right) \chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast},r\right) |Z_{i}^{\ast}\right] -\mathbb{E}^{\ast }\left[ \mathcal{K}_{\sigma_{N}}\left( X_{im}^{\ast\prime}\hat{\beta }\right) \chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast},r\right) \right] \right\} \nonumber\\ & +\frac{1}{N\left( N-1\right) }\sum_{i=1}^{N}\sum_{m=1,m\neq i} ^{N}\left\{ \mathcal{K}_{\sigma_{N}}\left( X_{im}^{\ast\prime}\hat{\beta }\right) \chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast},r\right) -\mathbb{E} ^{\ast}\left[ \mathcal{K}_{\sigma_{N}}\left( X_{im}^{\ast\prime}\hat{\beta }\right) \chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast},r\right) |Z_{i}^{\ast }\right] \right. \nonumber\\ & \left. -\mathbb{E}^{\ast}\left[ \mathcal{K}_{\sigma_{N}}\left( X_{im}^{\ast\prime}\hat{\beta}\right) \chi_{2}\left( Z_{i}^{\ast} ,Z_{m}^{\ast},r\right) |Z_{m}^{\ast}\right] +\mathbb{E}^{\ast}\left[ \mathcal{K}_{\sigma_{N}}\left( X_{im}^{\ast\prime}\hat{\beta}\right) \chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast},r\right) \right] \right\} .\nonumber\\ \equiv &\mathcal{T}_{N4,1}^{\ast}+\mathcal{T}_{N4,2}^{\ast}+\mathcal{T} _{N4,3}^{\ast}. \end{align} We analyze the three terms in ((ref)) one by one. For $\mathcal{T}_{N4,1}^{\ast},$ we have \begin{align} \mathcal{T}_{N4,1}^{\ast} & =\mathbb{E}^{\ast}\left[ \mathcal{K} _{\sigma_{N}}\left( X_{im}^{\ast\prime}\hat{\beta}\right) \chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast},r\right) \right] \\ & =\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{m=1}^{N}\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime}\hat{\beta}\right) \chi_{2}\left( Z_{i},Z_{m},r\right) \nonumber\\ & =\frac{1}{2}\left( r-\gamma\right) ^{\prime}\mathbb{V}_{\gamma}\left( r-\gamma\right) +2\left( r-\gamma\right) ^{\prime}\frac{W_{N2}}{\sqrt{N} }-2\left( r-\gamma\right) ^{\prime}\Gamma\mathbb{V}_{\beta}^{-1}\frac {W_{N1}}{\sqrt{N}}+o_{p}\left( N^{-1}\right) ,\nonumber \end{align} where the last line holds because the term in the second line is $2\left[ N\left( N-1\right) \right] ^{-1}\mathbb{L}_{N,\gamma}^{K}( r,\hat{\beta}) $ plus a small order term. For $\mathcal{T}_{N4,2}^{\ast},$ we first note that \begin{align} & \sum_{i=1}^{N}\left\{ \mathbb{E}\left[ \mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime}\beta\right) \chi_{2}\left( Z_{i},Z_{m},r\right) |Z_{i}\right] -\mathbb{E}\left[ \mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime}\beta\right) \chi_{2}\left( Z_{i},Z_{m},r\right) \right] \right\} \\ = &\sqrt{N}\left( r-\gamma\right) ^{\prime}W_{N2}+o_{p}\left( 1\right) ,\nonumber \end{align} for $\left\Vert r-\gamma\right\Vert =O\left( N^{-1/2}\right) ,$ which holds for the same reason as for equation ((ref))$.$ Since $\hat{\beta} -\beta=O_{p}\left( N^{-1/2}\right) ,$ by the equicontinuity we claimed, \begin{align} \sum_{i=1}^{N} & \left\{ \mathbb{E}\left[ \mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime}\beta\right) \chi_{2}\left( Z_{i},Z_{m},r\right) |Z_{i}\right] -\mathbb{E}\left[ \mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime}\beta\right) \chi_{2}\left( Z_{i},Z_{m},r\right) \right] \right\} \\ & -\sum_{i=1}^{N}\left\{ \mathbb{E}_{N}\left[ \mathcal{K}_{\sigma_{N} }\left( X_{im}^{\prime}\hat{\beta}\right) \chi_{2}\left( Z_{i} ,Z_{m},r\right) |Z_{i}\right] -\mathbb{E}_{N}\left[ \mathcal{K}_{\sigma _{N}}\left( X_{im}^{\prime}\hat{\beta}\right) \chi_{2}\left( Z_{i} ,Z_{m},r\right) \right] \right\} =o_{p}\left( 1\right) .\nonumber \end{align} Then equations ((ref)) and ((ref)) imply that \begin{align*} & 2\sum_{i=1}^{N}\left\{ \mathbb{E}_{N}\left[ \mathcal{K}_{\sigma_{N} }\left( X_{im}^{\prime}\hat{\beta}\right) \chi_{2}\left( Z_{i} ,Z_{m},r\right) |Z_{i}\right] -\mathbb{E}_{N}\left[ \mathcal{K}_{\sigma _{N}}\left( X_{im}^{\prime}\hat{\beta}\right) \chi_{2}\left( Z_{i} ,Z_{m},r\right) \right] \right\} \\ & =2\sqrt{N}\left( r-\gamma\right) ^{\prime}W_{N2}+o_{p}\left( 1\right) , \end{align*} which approximates a normal distribution for $\left\Vert r-\gamma\right\Vert =O\left( N^{-1/2}\right) .$ Thus, we are able to apply the results of gine1990bootstrapping and get that \[ N\mathcal{T}_{N4,2}^{\ast}=2\sum_{i=1}^{N}\left\{ \mathbb{E}^{\ast}\left[ \mathcal{K}_{\sigma_{N}}\left( X_{im}^{\ast\prime}\hat{\beta}\right) \chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast},r\right) |Z_{i}^{\ast}\right] -\mathbb{E}^{\ast}\left[ \mathcal{K}_{\sigma_{N}}\left( X_{im}^{\ast\prime }\hat{\beta}\right) \chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast},r\right) \right] \right\} \] approximates the distribution of the above term. That is \begin{equation} \mathcal{T}_{N4,2}^{\ast}=2\left( r-\gamma\right) ^{\prime}\frac {W_{N2}^{\ast}}{\sqrt{N}}+o_{p}\left( N^{-1}\right) , \end{equation} where $W_{N2}^{\ast}$ is an independent copy of $W_{N2}.$ The same arguments as we used for $\mathcal{T}_{N3}^{\ast}$ lead to \begin{equation} \mathcal{T}_{N4,3}^{\ast}=o_{p}\left( N^{-1}\right) . \end{equation} Then, equations ((ref)), ((ref)), ((ref) ), and ((ref)) together imply that for $\Vert r-\gamma\Vert=O ( N^{-1/2}) $, \begin{align} \mathcal{T}_{N4}^{\ast} = &\frac{1}{2}\left( r-\gamma\right) ^{\prime }\mathbb{V}_{\gamma}\left( r-\gamma\right) +2\left( r-\gamma\right) ^{\prime}\frac{W_{N2}}{\sqrt{N}}-2\left( r-\gamma\right) ^{\prime} \Gamma\mathbb{V}_{\beta}^{-1}\frac{W_{N1}}{\sqrt{N}}\\ & +2\left( r-\gamma\right) ^{\prime}\frac{W_{N2}^{\ast}}{\sqrt{N}} +o_{p}\left( N^{-1}\right) .\nonumber \end{align} For $\mathcal{T}_{N5}^{\ast},$ we have \begin{align*} \mathcal{T}_{N5}^{\ast} & =\left[ N\left( N-1\right) \right] ^{-1} \sum_{i=1}^{N}\sum_{m=1,m\neq i}^{N}\chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast },r\right) \nabla\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\ast\prime} \hat{\beta}\right) \left( \frac{X_{im}^{\ast\prime}\left( \hat{\beta} ^{\ast}-\hat{\beta}\right) }{\sigma_{N}}\right) \\ & =\left\{ \left[ N\left( N-1\right) \right] ^{-1}\sum_{i=1}^{N} \sum_{m=1,m\neq i}^{N}\chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast},r\right) \sigma_{N}^{-1}\nabla\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\ast\prime} \hat{\beta}\right) X_{im}^{\ast\prime}\right\} \left( \hat{\beta}^{\ast }-\hat{\beta}\right) . \end{align*} Following the same analysis for $\mathcal{T}_{N5},$ the lead term for $\mathcal{T} _{N5}^{\ast}$ is \[ \mathbb{E}^{\ast}\left[ \chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast},r\right) \sigma_{N}^{-1}\nabla\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\ast\prime} \hat{\beta}\right) X_{im}^{\ast\prime}\right] \left( \hat{\beta}^{\ast }-\hat{\beta}\right) . \] Note that \begin{align*} & \mathbb{E}^{\ast}\left[ \chi_{2}\left( Z_{i}^{\ast},Z_{m}^{\ast },r\right) \sigma_{N}^{-1}\nabla\mathcal{K}_{\sigma_{N}}\left( X_{im} ^{\ast\prime}\hat{\beta}\right) X_{im}^{\ast\prime}\right] \left( \hat{\beta}^{\ast}-\hat{\beta}\right) \\ & =\left[ \frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{m=1}^{N}\chi_{2}\left( Z_{i},Z_{m},r\right) \sigma_{N}^{-1}\nabla\mathcal{K}_{\sigma_{N}}\left( X_{im}^{\prime}\beta\right) X_{im}\right] \left( \hat{\beta}^{\ast} -\hat{\beta}\right) \\ & =\left( r-\gamma\right) ^{\prime}\Gamma\left( \hat{\beta}^{\ast} -\hat{\beta}\right) +O_{p}\left( \left\Vert r-\gamma\right\Vert ^{2}\left\Vert \hat{\beta}^{\ast}-\hat{\beta}\right\Vert \right) +o_{p}\left( N^{-1}\right) \\ & =-2\left( r-\gamma\right) ^{\prime}\Gamma\mathbb{V}_{\beta}^{-1} \frac{W_{N1}^{\ast}}{\sqrt{N}}+o_{p}\left( \frac{\left\Vert r-\gamma \right\Vert }{\sqrt{N}}\right) +O_{p}\left( \left\Vert r-\gamma\right\Vert ^{2}\left\Vert \hat{\beta}^{\ast}-\hat{\beta}\right\Vert \right) +o_{p}\left( N^{-1}\right) , \end{align*} where we use equations ((ref)) and ((ref)) to get the third line, and we substitute equation ((ref)) in the last line. Then we have \begin{equation} \mathcal{T}_{N5}^{\ast}=-2\left( r-\gamma\right) ^{\prime}\Gamma \mathbb{V}_{\beta}^{-1}\frac{W_{N1}^{\ast}}{\sqrt{N}}+o_{p}\left( \frac{\left\Vert r-\gamma\right\Vert }{\sqrt{N}}\right) +O_{p}\left( \left\Vert r-\gamma\right\Vert ^{2}\left\Vert \hat{\beta}^{\ast}-\hat{\beta }\right\Vert \right) +o_{p}\left( N^{-1}\right) . \end{equation} For $\mathcal{T}_{N6}^{\ast},$ it is not hard to see that \begin{equation} \mathcal{T}_{N6}^{\ast}=O_{p}\left( \frac{\left\Vert r-\gamma\right\Vert \left\Vert \hat{\beta}^{\ast}-\hat{\beta}\right\Vert ^{2}}{\sigma_{N}^{2} }\right) . \end{equation} Then, equations ((ref)), ((ref)), ((ref)), and ((ref)) together imply that for $\Vert r-\gamma\Vert=O\left( N^{-1/2}\right) $, \begin{align*} & 2\left[ N\left( N-1\right) \right] ^{-1}\mathbb{L}_{N,\gamma}^{K\ast }\left( r,\hat{\beta}^{\ast}\right) \\ =&\frac{1}{2}\left( r-\gamma\right) ^{\prime}\mathbb{V}_{\gamma}\left( r-\gamma\right) +2\left( r-\gamma\right) ^{\prime}\frac{W_{N2}}{\sqrt{N} }-2\left( r-\gamma\right) ^{\prime}\Gamma\mathbb{V}_{\beta}^{-1}\frac {W_{N1}}{\sqrt{N}}+2\left( r-\gamma\right) ^{\prime}\frac{W_{N2}^{\ast} }{\sqrt{N}}\\ & -2\left( r-\gamma\right) ^{\prime}\Gamma\mathbb{V}_{\beta}^{-1} \frac{W_{N1}^{\ast}}{\sqrt{N}}+o_{p}\left( N^{-1}\right), \end{align*} from which we can show that $\hat{\gamma}^{\ast}-\gamma=O_{p}\left( N^{-1/2}\right) .$ With this result and by the CMT, we get \[ \sqrt{N}\left( \hat{\gamma}^{\ast}-\gamma\right) =-2\mathbb{V}_{\gamma} ^{-1}W_{N2}+2\mathbb{V}_{\gamma}^{-1}\Gamma\mathbb{V}_{\beta}^{-1} W_{N1}-2\mathbb{V}_{\gamma}^{-1}W_{N2}^{\ast}+2\mathbb{V}_{\gamma}^{-1} \Gamma\mathbb{V}_{\beta}^{-1}W_{N1}^{\ast}+o_{p}\left( 1\right), \] which implies \[ \sqrt{N}\left( \hat{\gamma}^{\ast}-\hat{\gamma}\right) =-2\mathbb{V} _{\gamma}^{-1}W_{N2}^{\ast}+2\mathbb{V}_{\gamma}^{-1}\Gamma\mathbb{V}_{\beta }^{-1}W_{N1}^{\ast}+o_{p}\left( N^{-1}\right) \] by substituting equation ((ref)) in. Since $W_{N1}^{\ast}$ and $W_{N2}^{\ast}$ are independent copies of $W_{N1}$ and $W_{N2},$ respectively, $\sqrt{N}\left( \hat{\gamma}^{\ast}-\hat{\gamma }\right) $ approximates the distribution of $\sqrt{N}\left( \hat{\gamma }-\gamma\right) .$ \section{Additional Discussions} We discuss the convergence rates of our cross-sectional MRC and panel data MS estimators in Section (ref). We explain why we choose not to construct the panel data MS estimator for $\gamma$ via index matching in Section (ref). \subsection{On the Convergence Rates of MRC and MS Estimators} The rate of convergence of the MRC estimators for the cross-sectional model is $N^{-1/2}$. The intuition is that each observation can be matched with $Nh_{N}^{k_{1}+k_{2}}$ and $N\sigma_{N}^{2k_{1}}$ observations (in probability) for estimating $\beta$ and $\gamma$, respectively. Thus, all observations are useful and the curse of dimensionality is not a concern for the cross-sectional estimators. In contrast, an observation is useful for the panel data MS estimators if all its covariates are very close to each other across two time periods. For a fixed $T,$ the probability of an observation being useful is proportional to $h_{N}^{k_{1}+k_{2}}$ or $\sigma_{N}^{2k_{1}}.$ Therefore, the number of \textquotedblleft useful\textquotedblright\ observations is proportional to $Nh_{N}^{k_{1}+k_{2}}$ and $N\sigma_{N}^{2k_{1}},$ and the estimation suffers from the curse of dimensionality. We may remedy this problem if $T\rightarrow \infty$ such that $Th_{N}^{k_{1}+k_{2}}\rightarrow\infty$ and $T\sigma _{N}^{2k_{1}}\rightarrow\infty$ and each observation can be “matched” at certain two time periods with large probability. Since this paper focuses on models for short panel data, we leave the discussion on this issue to future research. Further, the criterion functions have a \textquotedblleft sharp\textquotedblright\ edge as in KimPollard1990. As a result, the convergence rates for $\hat{\beta}$ and $\hat{\gamma}$ are expected to be $(Nh_{N}^{k_{1}+k_{2}})^{-1/3}$ and $(N\sigma_{N}^{2k_{1}})^{-1/3}$, respectively. The cross-sectional estimators, however, do not suffer this \textquotedblleft sharp\textquotedblright\ edge effect because the criterion functions are U-statistics and the \textquotedblleft sharp\textquotedblright \ edge effect vanishes after we decompose the criterion functions (see, e.g., the second line of equation ((ref))) when deriving the asymptotics. \subsection{On the Construction of Panel Data MS Estimator } Unlike the cross-section case, our panel data MS procedure does not estimate $\gamma$ through matching $X_{jt}^{\prime}\hat{\beta}$ and $X_{js} ^{\prime}\hat{\beta}$, $j=1,2$, because of the following observation. As demonstrated in Theorem (ref), the convergence rate of $\hat{\beta}$ is $(Nh_{N}^{k_{1}+k_{2}})^{-1/3}$. If we knew the true value of $\beta$, $\hat{\gamma}$ could be obtained by matching $X_{jt}^{\prime}\beta$ and $X_{js}^{\prime}\beta$. Using the same arguments, we can show that this infeasible estimator has a rate of convergence $(N\sigma_{N}^{2})^{-1/3}$. As a result, it is only possibly beneficial to match $X_{jt}\hat{\beta}$ and $X_{js}\hat{\beta}$ for estimating $\gamma$ when $\sigma_{N}^{2} =o(h_{N}^{k_{1}+k_{2}})$; otherwise the asymptotics of $\hat{\beta}$ would dominate the limiting distribution of $\hat{\gamma}$. However, this involves careful selection of tuning parameters $h_{N}$ and $\sigma_{N}$. In more general cases, this selection relies not only on the dimension of the regressor space but also on the cardinality of the choice set, and thus should be made on a case-by-case basis. Because of the lack of a universally applicable treatment, we choose to simply match the covariates to avoid this complexity and the ensuing discussion. \subsection{Identification of the Case with Three Alternatives } This section discusses how to create identification inequalities for models with three stand-alone alternatives and four potential bundles. More complex cases can be handled similarly. Given these inequalities, one can prove the identification and construct estimation procedures for model coefficients using the same methods employed in Sections (ref) and (ref). The technical conditions required for identification are similar to those of the two-alternative model studied in Sections (ref) and (ref). We omit them for conciseness and only focus on the identification strategies. Throughout this section, we consider choice set $\mathcal{J}=\{0,1,2,3,(1,2),(1,3),(2,3),\left( 1,2,3\right) \}$ or equivalently $\mathcal{D}=\{d|d=(d_{1},d_{2},d_{3})\in\{0,1\}^{3}\}$, which is the set of all possible $d=(d_{1},d_{2},d_{3})\in\{0,1\}^{3}$. \subsubsection{Cross-Sectional MRC Method} In the cross-sectional model, assume that an agent chooses $d$ that maximizes the latent utility \begin{align} U_{d}=& \sum_{j=1}^{3}F_{j}(X_{j}^{\prime }\beta ,\epsilon _{j})\cdot d_{j}+ \eta _{110}\cdot F_{110}\left(W_{1}^{\prime }\gamma _{1}\right) \cdot d_{1}\cdot d_{2}+ \eta _{101}\cdot F_{101}\left(W_{2}^{\prime }\gamma _{2}\right) \cdot d_{1}\cdot d_{3} \nonumber \\ & +\eta _{011}\cdot F_{011}\left( W_{3}^{\prime }\gamma _{3}\right)\cdot d_{2}\cdot d_{3} + \eta _{111}\cdot F_{111}\left(W_{4}^{\prime }\gamma _{4}\right) \cdot d_{1}\cdot d_{2}\cdot d_{3}, \end{align} where $X_{1},X_{2},X_{3}\in \mathbb{R}^{k_{1}},\ W_{1}\in \mathbb{R} ^{k_{2}},W_{2}\in \mathbb{R}^{k_{3}},W_{3}\in \mathbb{R}^{k_{4}},W_{4}\in \mathbb{R}^{k_{5}}$, $\eta \equiv (\eta _{110},\eta _{101},\eta _{011},\eta _{111})\in \mathbb{R}_{+}^{4}$ are the bundle-specific unobserved heterogeneity, and all $F_j(\cdot)$'s and $F_d(\cdot)$'s are strictly increasing in all arguments. Note that by properly re-organizing the covariate vector, expression ((ref)) can accommodate both choice-specific and common regressors. See CameronTrivedi2005 p.498 for a more detailed discussion. Thus, for notational convenience, here we do not distinguish common regressors form those choice-specific regressors as we did in Section (ref). Let $Z\equiv (X_{1} ,X_{2} ,X_{3} ,W_{1} ,W_{2} ,W_{3} ,W_{4} ) $ to collect all observed covariates. We assume $(\epsilon _{1},\epsilon _{2},\epsilon _{3},\eta )\perp Z$. The observed dependent variable $Y_{d}$ takes the form \begin{equation*} Y_{d}=1[U_{d}>U_{d^{\prime}},\forall d^{\prime}\in\mathcal{D}\setminus d]. \end{equation*} Then, similar to the two-alternatives\footnote{This refers to models with $J=2$ or equivalently $\mathcal{J}=\{0,1,2,(1,2)\}$ that we discussed in Sections (ref) and (ref).} case, by $(\epsilon_{1},\epsilon_{2} \epsilon_{3},\eta)\perp Z,$ \begin{equation*} P(Y_{(1,d_{2},d_{3})}=1|Z,X_{2}=x_{2},X_{3}=x_{3},W_{1}=w_{1},W_{2}=w_{2},W_{3}=w_{3},W_{4}=w_{4}) \end{equation*} is increasing in $X_{1}^{\prime}\beta$ for any constant vectors $\left( x_{2},x_{3},w_{1},w_{2},w_{3},w_{4}\right)$ and any $\left( d_{2},d_{3}\right) $. Let $\left( Y_{i},Z_{i}\right) $ and $\left( Y_{m},Z_{m}\right) $ be two independent copies of $\left( Y,Z\right) $ with \begin{equation*} Y\equiv(Y_{(0,0,0)},Y_{(1,0,0)},Y_{(0,1,0)},Y_{(0,0,1)},Y_{(1,1,0)},Y_{(1,0,1)},Y_{(0,1,1)},Y_{(1,1,1)}). \end{equation*} Then we have the following (conditional) moment inequalities for all $d_{2}$ and $d_{3}$: \begin{align} & \left\{ X_{i1}^{\prime}\beta\geq X_{m1}^{\prime}\beta\right\} \\ \Leftrightarrow & \left\{ P(Y_{i\left( 1,d_{2},d_{3}\right) }=1|Z_{i},X_{i2}=x_{2},X_{i3}=x_{3},W_{i1}=w_{1},W_{i2}=w_{2},W_{i3}=w_{3},W_{i4}=w_{4})\right. \notag \\ & \geq\left. P(Y_{m\left( 1,d_{2},d_{3}\right) }=1|Z_{m},X_{m2}=x_{2},X_{m3}=x_{3},W_{m1}=w_{1},W_{m2}=w_{2},W_{m3}=w_{3},W_{m4}=w_{4})\right\} . \notag \end{align} Similar moment inequalities can be obtained for $X_{ij}^{\prime}\beta\geq X_{mj}^{\prime}\beta$ by fixing $\left\{ X_{1},X_{2},X_{3}\right\} \backslash\left\{ X_{j}\right\} $ and $W_{1},W_{2},W_{3},W_{4}$. Collectively, these moment inequalities establish the identification of $ \beta,$ with some standard technical conditions as for the two-alternatives case. Once $\beta $ is identified, we can proceed to identify $\gamma $'s by fixing $X_{j}^{\prime }\beta ,$ $j=1,2,3,$ , or simply by fixing $X_{1},X_{2},X_{3}$. We follow the former strategy. For $\gamma _{1},$ we fix $(X_{1}^{\prime }\beta ,X_{2}^{\prime }\beta ,X_{3}^{\prime }\beta )$ at some constant vector $(v_{1},v_{2},v_{3})$ and fix $(W_{i2},W_{i3},W_{i4})$ at some constant vector $(w_2,w_3,w_4)$. Then by $(\epsilon _{1},\epsilon _{2},\epsilon _{3},\eta )\perp Z,$ \begin{equation*} P(Y_{(1,1,d_{3})}=1|Z,X_{1}^{\prime }\beta =v_{1},X_{2}^{\prime }\beta =v_{2},X_{3}^{\prime }\beta =v_{3},W_{2}=w_{2},W_{3}=w_{3},W_{4}=w_{4}) \end{equation*} is increasing in $W_{1}^{\prime }\gamma_1 $ for any $d_{3}.$ As a result, for any $d_{3}$, $(v_{1},v_{2},v_{3})$, and $(w_{2},w_{3},w_{4})$, we have \begin{align} & \left\{ W_{i1}^{\prime }\gamma _{1}\geq W_{m1}^{\prime }\gamma _{1}\right\} \\ \Leftrightarrow &\left\{ P(Y_{i(1,1,d_{3})}=1|Z_{i},X_{i1}^{\prime }\beta =v_{1},X_{i2}^{\prime }\beta =v_{2},X_{i3}^{\prime }\beta =v_{3},W_{i2}=w_{2},W_{i3}=w_{3},W_{i4}=w_{4})\right. \notag \\ & \geq \left. P(Y_{m(1,1,d_{3})}=1|Z_{m},X_{m1}^{\prime }\beta =v_{1},X_{m2}^{\prime }\beta =v_{2},X_{m3}^{\prime }\beta =v_{3},W_{m2}=w_{2},W_{m3}=w_{3},W_{m4}=w_{4})\right\} . \notag \end{align} We can similarly identify $\gamma _{2}$ and $\gamma _{3}.$ Once $\beta$, $ \gamma_{1},\gamma_{2},$ and $\gamma_{3}$ are identified, we can identify $\gamma_{4}$ by fixing either $( X_{1},X_{2},X_{3},W_{1},W_{2},W_{3})$ or $\left( X_{1}^{\prime}\beta,X_{2}^{\prime}\beta ,X_{3}^{\prime}\beta,W_{1}^{\prime}\gamma_{1},W_{2}^{\prime} \gamma_{2},W_{3}^{\prime}\gamma_{3}\right)$. We consider the latter and fix $(X_{1}^{\prime}\beta,X_{2}^{\prime}\beta,X_{3}^{ \prime}\beta ,W_{1}^{\prime}\gamma_{1},W_{2}^{\prime}\gamma_{2},W_{3}^{\prime}\gamma _{3})$ at some constant vector $(v_{1},v_{2},v_{3},v_{4},v_{5},v_{6})$. Then $ (\epsilon_{1},\epsilon_{2},\epsilon_{3},\eta)\perp Z$ implies that \begin{equation*} P\left( Y_{(1,1,1)}=1|Z,X_{1}^{\prime}\beta=v_{1},X_{2}^{\prime}\beta =v_{2},X_{3}^{\prime}\beta=v_{3},W_{1}^{\prime}\gamma_{1}=v_{4},W_{2}^{ \prime }\gamma_{2}=v_{5},W_{3}^{\prime}\gamma_{3}=v_{6}\right) \end{equation*} is increasing in $W_{4}^{\prime}\gamma_{4}$ for any fixed $ (v_{1},v_{2},v_{3},v_{4},v_{5},v_{6})$. Thus, the following inequality holds for any $(v_{1},v_{2},v_{3},v_{4},v_{5},v_{6})^{\prime},$ \begin{align} & \left\{ W_{i4}^{\prime}\gamma_{4}\geq W_{m4}^{\prime}\gamma_{4}\right\} \\ \Leftrightarrow &\left\{ P(Y_{i(1,1,1)}=1|Z_{i},X_{i1}^{\prime}\beta =v_{1},X_{i2}^{\prime}\beta=v_{2},X_{i3}^{\prime}\beta=v_{3},W_{i1}^{\prime }\gamma_{1}=v_{4},W_{i2}^{\prime}\gamma_{2}=v_{5},W_{i3}^{\prime}\gamma _{3}=v_{6})\right. \notag \\ & \geq\left. P(Y_{m(1,1,1)}=1|Z_{m},X_{m1}^{\prime}\beta=v_{1},X_{m2}^{\prime} \beta=v_{2},X_{m3}^{\prime}\beta=v_{3},W_{m1}^{\prime} \gamma_{1}=v_{4},W_{m2}^{\prime}\gamma_{2}=v_{5},W_{m3}^{\prime}\gamma _{3}=v_{6})\right\} . \notag \end{align} The inequalities established in equations ((ref))--((ref)) can be used to construct localized MRC estimators as in Section (ref). However, this matching-based identification strategy cannot identify coefficients on common regressors, i.e., covariates that appear in more than one index. \subsubsection{LAD Method} The LAD estimation for $\theta:=(\beta,\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4})$ is analogous to the simple case presented in the main text. Here, we only outline the procedure for the cross-sectional model for illustration purposes. The panel data LAD estimation can be constructed similarly and is omitted for brevity.\footnote{For the panel data setting, the identification relies on the conditional homogeneity condition $\xi_{s}\overset{d}{=}\xi_{t}|(\alpha,Z_{s},Z_{t})$.} \begin{enumerate} • By $(\epsilon_{1},\epsilon_{2},\epsilon_{3},\eta)\perp Z$ and the monotonicity of $F_{j}(\cdot)$'s and $F_{d}(\cdot)$'s functions, we can write \begin{align} \{X_{im1}'\beta\geq0,X_{im2}'\beta\leq0,X_{im3}'\beta\leq0, & W_{im1}'\gamma_{1}\leq0,W_{im2}'\gamma_{2}\leq0,W_{im3}'\gamma_{3},\leq0,W_{im4}'\gamma_{4}\leq0\}\\ & \Rightarrow\nonumber \\ \Delta p_{(1,0,0)}(Z_{i},Z_{m}):=P(Y_{i(1,0,0)} & =1|Z_{i})-P(Y_{m(1,0,0)}=1|Z_{i})\geq0 \end{align} and \begin{align} \{X_{im1}'\beta\leq0,X_{im2}'\beta\geq0,X_{im3}'\beta\geq0, & W_{im1}'\gamma_{1}\geq0,W_{im2}'\gamma_{2}\geq0,W_{im3}'\gamma_{3},\geq0,W_{im4}'\gamma_{4}\geq0\}\\ & \Rightarrow\nonumber \\ \Delta p_{(1,0,0)}(Z_{i},Z_{m}) & \leq0, \end{align} i.e., the event ((ref)) and ((ref)) are respectively sufficient (but not necessary) conditions for inequality ((ref)) and ((ref)) to hold. • Define \[ Q_{(1,0,0)}(\vartheta):=\mathbb{E}[q_{(1,0,0)}(Z_{i},Z_{m},\vartheta)], \] where \begin{align*} & q_{(1,0,0)}(Z_{i},Z_{m},\vartheta)\\ = & [\vert I_{im(1,0,0)}^{+}(\vartheta)-1[\Delta p_{(1,0,0)}(Z_{i},Z_{m})\geq0]\vert+\vert I_{im(1,0,0)}^{-}(\vartheta)-1[\Delta p_{(1,0,0)}(Z_{i},Z_{m})\leq0]\vert]\\ & \times[I_{im(1,0,0)}^{+}(\vartheta)+I_{im(1,0,0)}^{-}(\vartheta)] \end{align*} with \begin{align*} & I_{im(1,0,0)}^{+}(\vartheta)\\ = & 1[X_{im1}'\beta\geq0,X_{im2}'\beta\leq0,X_{im3}'\beta\leq0,W_{im1}'\gamma_{1}\leq0,W_{im2}'\gamma_{2}\leq0,W_{im3}'\gamma_{3},\leq0,W_{im4}'\gamma_{4}\leq0]\\ & I_{im(1,0,0)}^{-}(\vartheta)\\ = & 1[X_{im1}'\beta\leq0,X_{im2}'\beta\geq0,X_{im3}'\beta\geq0,W_{im1}'\gamma_{1}\geq0,W_{im2}'\gamma_{2}\geq0,W_{im3}'\gamma_{3},\geq0,W_{im4}'\gamma_{4}\geq0]. \end{align*} The intuition underlying the construction of this population criterion function is the same as the discussion in Section (ref) (below ((ref))--((ref))). Then the identification of $\theta$ can be established using arguments similar to those for Theorem (ref). • To compute the estimator $\hat{\theta}$ for $\theta$, one can consider the “debiased” sample criterion analogous to ((ref)), i.e., \begin{equation} \hat{\theta}=\arg\min_{\vartheta\in\Theta}\sum_{i=1}^{N-1}\sum_{m>i}\hat {q}^{D}_{im(1,0,0)}(\vartheta), \end{equation} where \begin{align*} \hat{q}_{im(1,0,0)}^{D}(\vartheta)= & [\vert I_{im(1,0,0)}^{+}(\vartheta)-\Delta \hat{p}_{(1,0,0)}(Z_{i},Z_{m})\vert+\vert I_{im(1,0,0)}^{-}(\vartheta)+\Delta\hat {p}_{(1,0,0)}(Z_{i},Z_{m})\vert]\\ & \times[I_{im(1,0,0)}^{+}(\vartheta)+I_{im(1,0,0)}^{-}(\vartheta)]+[1-(I_{im(1,0,0)} ^{+}(\vartheta)+I_{im(1,0,0)}^{-}(\vartheta))]. \end{align*} • Repeat Steps 1--3 to obtain sample criteria associated with $\Delta p_{(0,1,0)}(Z_{i},Z_{m})$, $\Delta p_{(0,0,1)}(Z_{i},Z_{m})$, $\Delta p_{(1,1,0)}(Z_{i},Z_{m})$, $\Delta p_{(1,0,1)}(Z_{i},Z_{m})$, $\Delta p_{(0,1,1)}(Z_{i},Z_{m})$, $\Delta p_{(0,0,0)}(Z_{i},Z_{m})$, and $\Delta p_{(1,1,1)}(Z_{i},Z_{m})$. Collectively, use some convex combination of these criteria to obtain $\hat{\theta}$ with better finite sample performance. \end{enumerate} \subsubsection{Panel Data MS Method} We specify the latent utility of the panel data bundle choice model with $J=3$ as follows: \begin{align*} U_{dt}=& \sum_{j=1}^{3}F_{j}(X_{jt}^{\prime }\beta ,\epsilon _{jt},\alpha _{j})\cdot d_{j}+\eta _{110t}\cdot F_{110}\left( W_{1t}^{\prime }\gamma _{1}+\alpha _{b1}\right) \cdot d_{1}\cdot d_{2}+\eta _{101t}\cdot F_{101}\left(W_{2t}^{\prime }\gamma _{2}+\alpha _{b2}\right) \cdot d_{1}\cdot d_{3} \\ & +\eta _{011t}\cdot F_{011}\left(W_{3t}^{\prime }\gamma _{3}+\alpha _{b3}\right) \cdot d_{2}\cdot d_{3}+\eta _{111t}\cdot F_{111}\left( W_{4t}^{\prime }\gamma _{4}+\alpha _{b4}\right) \cdot d_{1}\cdot d_{2}\cdot d_{3}, \end{align*} where $X_{1t},X_{2t},X_{3t}\in \mathbb{R}^{k_{1}},\ W_{1t}\in \mathbb{R} ^{k_{2}},W_{2t}\in \mathbb{R}^{k_{3}},W_{3t}\in \mathbb{R}^{k_{4}},$ $W_{4t}\in \mathbb{R}^{k_{5}}$, and $\eta _{t}\equiv (\eta _{110t},\eta _{101t},\eta _{011t},\eta _{111t})\in \mathbb{R}_{+}^{4}$. All $F_j(\cdot)$'s and $F_d(\cdot)$'s are strictly increasing in all arguments. An agent chooses $d$ that maximizes the latent utility, i.e., \begin{equation*} Y_{dt}=1[U_{dt}>U_{d^{\prime }t},\forall d^{\prime }\in \mathcal{D}\setminus d]. \end{equation*} Denote $Z_{t} =(X_{1t},X_{2t},X_{3t},W_{1t},W_{2t},W_{3t},W_{4t})$, $\xi _{t}=(\epsilon _{1t},\epsilon _{2t},\epsilon _{3t},\eta _{110t},\eta _{101t},\eta _{011t},\eta _{111t})$, and $\alpha =\left( \alpha _{1},\alpha _{2},\alpha _{3},\alpha _{b1},\alpha _{b2},\alpha _{b3},\alpha _{b4}\right)$. The key identification assumption is that $\xi _{s}\overset{d}{=}\xi _{t}|(\alpha ,Z_{s},Z_{t})$ holds for any two time periods $s$ and $t$. The identification of the panel data model is similar to the cross-sectional model discussed in Section (ref). The primary difference is that in a panel data setting, we need to match and compare the same agent in time periods $s$ and $t$, rather than two independent agents $i$ and $m$. As a result, we can construct the same set of identification inequalities as Section (ref), but replace the $(i,m)$ therein with $(s,t)$. For example, analogous to ((ref)) and ((ref)), we can write for any $ d_{2},d_{3},x_{1},x_{2},x_{3},w_{1},w_{2},w_{3},w_{4}$, and $c$ \begin{align} & \left\{ X_{1t}^{\prime }\beta \geq X_{1s}^{\prime }\beta \right\} \\ \Leftrightarrow &\{P(Y_{\left( 1,d_{2},d_{3}\right)t }=1|Z_t,X_{2t}=x_{2},X_{3t}=x_{3},W_{1t}=w_{1},W_{2}=w_{2t},W_{3t}=w_{3},W_{4t}=w_{4},\alpha =c) \notag \\ & \geq P(Y_{\left( 1,d_{2},d_{3}\right)s }=1|Z_s,X_{2s}=x_{2},X_{3s}=x_{3},W_{1s}=w_{1},W_{2s}=w_{2},W_{3s}=w_{3},W_{4s}=w_{4},\alpha =c)\} \notag \end{align} and \begin{align} & \left\{W_{1t}^{\prime }\gamma _{1}\geq W_{1s}^{\prime }\gamma _{1}\right\} \\ \Leftrightarrow & P(Y_{\left( 1,1,d_{3}\right) t}=1|Z_t,X_{1t}=x_{1},X_{2t}=x_{2},X_{3t}=x_{3},W_{2t}=w_{2},W_{3t}=w_{3},W_{4t}=w_{4},\alpha =c) \notag \\ & \geq P(Y_{(1,1,d_{3})s}=1|Z_s,X_{1s}=x_{1},X_{2s}=x_{2},X_{3s}=x_{3},W_{2s}=w_{2},W_{3s}=w_{3},W_{4s}=w_{4},\alpha =c). \notag \end{align} Then, under similar regular conditions, we can identify and estimate $(\beta,\gamma_1)$ based on inequalities ((ref)) and ((ref)). Identification inequalities for $(\gamma_2,\gamma_3,\gamma_4)$ can be similarly constructed and hence are omitted for brevity. The panel data MS estimator differs from the cross-sectional MRC estimator in that its convergence rate decreases as the number of matching variables increases. For example, using similar analysis, we can show that for MS estimators based on inequalities ((ref)) and ((ref)), we have \begin{equation*} \hat{\beta}-\beta =O_{p}\left( (Nh_{N}^{2k_{1}+k_{2}+k_{3}+k_{4}+k_{5}})^{-1/3}\right) \text{ and }\hat{ \gamma}_{1}-\gamma _{1}=O_{p}\left( (N\sigma_{N}^{3k_{1}+k_{3}+k_{4}+k_{5}})^{-1/3}\right), \end{equation*} where we assume we use the same type of estimator as in Section (ref) and the same bandwidth ($h_{N}$ or $\sigma_{N}$) for all covariates. The convergence rates of $\hat{\gamma}_{2},\hat{\gamma}_{3}$, and $\hat{\gamma}_{4}$ can be similarly obtained. Generally speaking, we can show that the MS estimator has a convergence rate of $O_p((Nh_N^{\bar{k}})^{-1/3})$ ($O_p((N\sigma_N^{\bar{k}})^{-1/3})$), where $\bar{k}$ represents the number of matching variables. \section{Proofs of Technical Lemmas} We prove Lemmas (ref)--(ref) and Lemmas (ref)--(ref) in order. For Lemmas (ref)--(ref), we use the following results, which can be derived under Assumptions C2 and C5 by applying standard arguments (e.g., Example 6.4 of KimPollard1990). We use $\Vert \cdot \Vert _{F}$ to denote the Frobenius norm; that is, for any matrix $\mathbf{A}$, $\Vert \mathbf{A}\Vert _{F}=\sqrt{\text{trace}\left( \mathbf{A}\mathbf{A}^{\prime }\right) }$. Let $\mathcal{N}_{\beta }$ denote a neighborhood of $\beta$. Then, for $\varrho _{i}(b)$ defined in ((ref)), \begin{enumerate} • All mixed second partial derivatives of $\varrho _{i}(b)$ exist on $ \mathcal{N}_{\beta }$. • There is an integrable function $M_{1}(Z_{i})$ such that for all $ b\in \mathcal{N}_{\beta }$, $\Vert \nabla ^{2}\varrho _{i}(b)-\nabla ^{2}\varrho _{i}(\beta )\Vert _{F}\leq M_{1}(Z_{i})\Vert b-\beta \Vert $. • $\mathbb{E}[\Vert \nabla \varrho _{i}(\beta )\Vert ^{2}]<\infty $, $ \mathbb{E}[\Vert \nabla ^{2}\varrho _{i}(\beta )\Vert _{F}]<\infty $, and $ \mathbb{E}[\nabla ^{2}\varrho _{i}(\beta )]$ is negative definite. \end{enumerate} Let $\mathcal{N}_{\gamma }$ denote a neighborhood of $ \gamma$. Then, for $\tau _{i}(r)$ and $\mu (v_{1},v_{2},r)$ defined respectively in ((ref)) and ((ref)), \begin{enumerate} • All mixed second partial derivatives of $\tau _{i}(r)$ exist on $ \mathcal{N}_{\gamma }$. • There is an integrable function $M_{2}(Z_{i})$ such that for all $ r\in \mathcal{N}_{\gamma }$ $\Vert \nabla ^{2}\tau _{i}(r)-\nabla ^{2}\tau _{i}(\gamma )\Vert _{F}\leq M_{2}(Z_{i})\Vert r-\gamma \Vert $. • $\mathbb{E}[\Vert \nabla \tau _{i}(\gamma )\Vert ^{2}]<\infty $, $ \mathbb{E}[\Vert \nabla ^{2}\tau _{i}(\gamma )\Vert _{F}]<\infty $, and $ \mathbb{E}[\nabla ^{2}\tau _{i}(\gamma )]$ is negative definite. • All mixed second partial derivatives of $\mu (v_{1},v_{2},r)$ exist on $\mathcal{N}_{\gamma }$. • There exists an continuous, integrable function $M_{3}(v_{1},v_{2})$ such that for all $r\in \mathcal{N}_{\gamma }$, $\Vert \nabla _{11}^{2}\mu (v_{1},v_{2},r)-\nabla _{11}^{2}\mu (v_{1},v_{2},\gamma )\Vert _{F}\leq M_{3}(v_{1},v_{2})\Vert r-\gamma \Vert $. • $\mathbb{E}[\Vert \nabla _{13}^{2}\mu (V_{i}\left( \beta \right) ,V_{i}\left( \beta \right) ,\gamma )\Vert _{F}^{2}]<\infty $ and $\sup_{r\in \mathcal{N}_{\gamma }}\mathbb{E}[\Vert \nabla _{33}^{2}\nabla _{1}\mu (V_{i}\left( \beta \right) ,V_{i}\left( \beta \right) ,\gamma )\Vert _{F}]<\infty $. \end{enumerate} Further, using the definition of $\tau _{i}(r)$, we can write $\tau _{i}(r)$ as \begin{equation*} \tau _{i}(r)=f_{V_{im}}(0)\mathbb{E} [h_{im}(r)|V_{i}=v_{i},W_{i}=w_{i},V_{im}=0]=\int B(v_{i},v_{i},w_{i},w_{m})S_{im}(r)f_{V,W}(v_{i},w_{m})dw_{m}. \end{equation*} \begin{proof} [Proof of Lemma (ref)]First, note that \begin{equation} \frac{1}{2}\sigma_{N}^{2}\sup_{r\in\mathcal{R}}|\hat{\mathcal{L}}_{N} ^{K}(r)-\mathcal{L}_{N}^{K}(r)|\leq\frac{1}{N(N-1)}\sum_{i\neq m} |K_{\sigma_{N},\gamma}(\hat{V}_{im})-K_{\sigma_{N},\gamma}(V_{im})|. \end{equation} Applying a second-order Taylor expansion to the right-hand side of ((ref)) yields the lead term of the form \begin{align*} & \frac{1}{\sigma_{N}N(N-1)}\sum_{i\neq m}|\nabla_{1}K_{\sigma_{N},\gamma }(V_{im})X_{im1}^{\prime}(\hat{\beta}-\beta)+\nabla_{2}K_{\sigma_{N},\gamma }(V_{im})X_{im2}^{\prime}(\hat{\beta}-\beta)|\\ \leq & \frac{1}{\sigma_{N}N(N-1)}\sum_{i\neq m}\sum_{j=1}^{2}|\nabla _{j}K_{\gamma}(V_{im}/\sigma_{N})X_{imj}^{\prime}(\hat{\beta}-\beta)|, \end{align*} where $\nabla_{j}K_{\gamma}(\cdot)$ ($\nabla_{j}K_{\sigma_{N},\gamma}(\cdot)$) denotes the partial derivative of $K_{\gamma}(\cdot)$ ($K_{\sigma_{N},\gamma }(\cdot)$) w.r.t. its $j$-th argument corresponding to alternative $j$. It suffices to focus on the term with $j=1$. Then by Assumptions C1--C7, and the Cauchy-Schwarz inequality, we conclude that \begin{align*} & \frac{1}{N(N-1)}\sum_{i\neq m}\left\vert \nabla_{1}K_{\gamma}(V_{im} /\sigma_{N})\frac{X_{im1}^{\prime}(\hat{\beta}-\beta)}{\sigma_{N}}\right\vert \leq\frac{1}{N(N-1)}\sum_{i\neq m}|\nabla_{1}K_{\gamma}(V_{im}/\sigma _{N})|\Vert X_{im1}\Vert\frac{\Vert\hat{\beta}-\beta\Vert}{\sigma_{N}}\\ = & [O_{p}(\sigma_{N}^{2})+o_{p}(1)]O_{p}(\Vert\hat{\beta}-\beta\Vert /\sigma_{N})=o_{p}(1), \end{align*} where the penultimate equality follows from the SLLN of U-statistics (see e.g., Serfling2009) and $\mathbb{E}[|\nabla_{1}K_{\gamma}(V_{im} /\sigma_{N})|\Vert X_{im1}\Vert]=O_{p}(\sigma_{N}^{2})$, and the last equality is due to the $\sqrt{N}$-consistency of $\hat{\beta}$. Then the desired convergence result follows. \end{proof} \begin{proof} [Proof of Lemma (ref)]Note that Lemma (ref) follows from $\sup_{r\in\mathcal{R}}|\mathcal{L}_{N}^{K}(r)-\mathbb{E}[\mathcal{L}_{N} ^{K}(r)]|=o_{p}(1)$ and $\sup_{r\in\mathcal{R}}|\mathbb{E}[\mathcal{L}_{N} ^{K}(r)]-\mathcal{L}(r)|=o(1)$. Define $\mathcal{F}_{N}=\{K_{\sigma_{N},\gamma}(v_{im})h_{im}(r)|r\in \mathcal{R}\}$ with $\sigma_{N}>0$ and $\sigma_{N}\rightarrow0$. $\mathcal{F}_{N}$ is a sub-class of the fixed class $\mathcal{F} \equiv\{K_{\gamma}(v_{im}/\sigma)h_{im}(r)|r\in\mathcal{R},\sigma >0\}=\mathcal{F}_{r}\mathcal{F}_{\sigma}$, with $\mathcal{F}_{\sigma} \equiv\{K_{\gamma}(v_{im}/\sigma)|\sigma>0\}$, which is Euclidean for the constant envelope $\sup_{v\in\mathbb{R}^{2}}|K_{\gamma}(v)|$ by Lemma 22 in NolanPollard1987 and $\mathcal{F}_{r}\equiv\{h_{im}(r)|r\in \mathcal{R}\}$. Noticing that $h_{im}(r)$ is uniformly bounded by 2, Example 2.11 and Lemma 2.15 in PakesPollard1989 then implies that $\mathcal{F}_{r}$ is Euclidean for the constant envelope 2. Putting all these results together, we conclude using Lemma 2.14 in PakesPollard1989 that $\mathcal{F}$ is Euclidean for the constant envelope $2\sup_{v\in \mathbb{R}^{2}}|K_{\gamma}(v)|$. Applying Corollary 4 in Sherman1994AoS , we obtain that $\sup_{r\in\mathcal{R}}|\mathcal{L}_{N}^{K}(r)-\mathbb{E} [\mathcal{L}_{N}^{K}(r)]|=O_{p}(N^{-1}/\sigma_{N}^{2})=o_{p}(1)$ by Assumption C7. It remains to show that $\sup_{r\in\mathcal{R}}|\mathbb{E}[\mathcal{L}_{N} ^{K}(r)]-\mathcal{L}(r)|=o(1)$. Let $\eta(\cdot)\equiv f_{V_{im}} (\cdot)\mathbb{E}[h_{im}(r)|V_{im}=\cdot]$. Then $\mathcal{L}(r)=\eta(0)$ by definition. We can write by Assumptions C6 and C7 that \begin{align*} & \sup_{r\in\mathcal{R}}|\mathbb{E}[\mathcal{L}_{N}^{K}(r)]-\mathcal{L} (r)|=\sup_{r\in\mathcal{R}}|\sigma_{N}^{-2}\mathbb{E}[K_{\gamma}(V_{im} /\sigma_{N})h_{im}(r)]-\eta(0)|\\ = & \sup_{r\in\mathcal{R}}|\int\sigma_{N}^{-2}K_{\gamma}(v/\sigma_{N} )\eta(v)\text{d}v-\eta(0)|=\sup_{r\in\mathcal{R}}|\int\sigma_{N}^{-2} K_{\gamma}(v/\sigma_{N})[\eta(0)+v^{\prime}\nabla\eta(\bar{v})]\text{d} v-\eta(0)|\\ = & \sup_{r\in\mathcal{R}}|\int K_{\gamma}(u)[\eta(0)+\sigma_{N}u^{\prime }\nabla\eta(u_{N})]\text{d}u-\eta(0)|=\sup_{r\in\mathcal{R}}|\sigma_{N}\int K_{\gamma}(u)u^{\prime}\nabla\eta(u_{N})\text{d}u|\\ \leq & \sigma_{N}\cdot\sup_{r\in\mathcal{R}}\int|K_{\gamma}(u)|\Vert u\Vert\Vert\nabla\eta(u_{N})\Vert\text{d}u=O(\sigma_{N})=o(1), \end{align*} where the third equality applies a mean-value expansion and the fourth equality uses a change of variables. Then, the desired result follows. \end{proof} \begin{proof} [Proof of Lemma (ref)]The proof proceeds by verifying the four sufficient conditions for Theorem 2.1 in NeweyMcFadden1994: (S1) $\mathcal{R}$ is a compact set, (S2) $\sup_{r\in\mathcal{R}}|\hat{\mathcal{L} }_{N}^{K}(r)-\mathcal{L}(r)|=o_{p}(1)$, (S3) $\mathcal{L}(r)$ is continuous in $r$, and (S4) $\mathcal{L}(r)$ is uniquely maximized at $\gamma$. Condition (S1) is satisfied by Assumption C4. Lemmas (ref) and (ref) together imply that condition (S2) holds. Note that \begin{align*} & \mathbb{E}[h_{im}(r)|V_{im}=0]= \mathbb{E}\left\{ \mathbb{E} [Y_{im(1,1)}(\text{sgn}(W_{im}^{\prime}r)-\text{sgn}(W_{im}^{\prime} \gamma))|Z_{i},Z_{m}]|V_{im}=0\right\} \\ = & \mathbb{E}\left[ (P(Y_{i(1,1)}=1|Z_{i})-P(Y_{m(1,1)}=1|Z_{m} ))(\text{sgn}(W_{im}^{\prime}r)-\text{sgn}(W_{im}^{\prime}\gamma ))|V_{im}=0\right] . \end{align*} Then the identification condition (S4) can be verified by similar arguments used in the proof of Theorem 2.1 given that $f_{V_{im} }(0)>0$ by Assumption C5. It remains to verify the continuity of $\mathcal{L}(r)$ in $r$. $\mathbb{E}[h_{im}(r)|V_{im}=0]$ can be expressed as the sum of terms like \begin{align*} & P(Y_{im(1,1)}=d,W_{im}^{(1)}>-\tilde{W}_{im}^{\prime}\tilde{r} |V_{im}=0)\\ = & \int\int_{-\tilde{W}_{im}^{\prime}\tilde{r}/r^{(1)}}^{\infty }P(Y_{im(1,1)}=d|W_{im}^{(1)}=w,\tilde{W}_{im},V_{im}=0)f_{W_{im}^{(1)} |\tilde{W}_{im},V_{im}=0}(w)\text{d}w\text{d}F_{\tilde{W}_{im}|V_{im}=0}. \end{align*} for some $d\in\{-1,0,1\}$. Then $\mathcal{L}(r)$ is continuous in $r$ if $f_{W_{im}^{(1)}|\tilde{W}_{im},V_{im}=0}(\cdot)$ does not have any mass points, which is guaranteed by Assumption C3. This completes the proof. \end{proof} \begin{proof} [Proof of Lemma (ref)]First, we express $\mathbb{E}[\mathcal{L} _{N}^{K}(r)]$ as the integral \begin{align} & \int\sigma_{N}^{-2}K_{\sigma_{N},\gamma}(v_{im})B(v_{i},v_{m},w_{i} ,w_{m})S_{im}(r)\text{d}F_{V,W}(v_{i},w_{i})\text{d}F_{V,W}(v_{m} ,w_{m})\\ = & \int K_{\gamma}(u_{im})B(v_{m}+u_{im}\sigma_{N},v_{m},w_{i},w_{m} )S_{im}(r)f_{V,W}(v_{m}+u_{im}\sigma_{N},w_{i})\text{d}w_{i}\text{d} u_{im}\text{d}F_{V,W}(v_{m},w_{m}),\nonumber \end{align} where we apply the change of variables $u_{im}=v_{im}/\sigma_{N}$ to obtain the equality. Take the $\kappa_{\gamma}^{\text{th}}$-order Taylor expansion inside the integral around $v_{m}$ to obtain the lead term \begin{align} & \int K_{\gamma}(u_{im})B(v_{m},v_{m},w_{i},w_{m})S_{im}(r)f_{V,W} (v_{m},w_{i})\text{d}w_{i}\text{d}u_{im}\text{d}F_{V,W}(v_{m},w_{m} )\nonumber\\ = & \int B(v_{m},v_{m},w_{i},w_{m})S_{im}(r)f_{V,W}(v_{m},w_{i} )\text{d}w_{i}\text{d}F_{V,W}(v_{m},w_{m})=\mathbb{E}[\tau_{m}(r)], \end{align} where the first equality follows by $\int K_{\gamma}(u)du=1$.\footnote{More precisely, we apply $\kappa_{1\gamma}^{\text{th}}$-order and $\kappa_{2\gamma}^{\text{th}}$-order Taylor expansions to $B(\cdot,v_{m},w_{i},w_{m} )$ and $f_{V,W}(\cdot,w_{i})$, respectively.} All remaining terms are zero except the last one which is of order $O(\sigma_{N}^{\kappa_\gamma }\delta_{N})=o(\Vert r-\gamma\Vert^{2})$ by Assumptions C6 and C7. Note that a second-order Taylor expansion around $\gamma$ gives \begin{align*} & \tau_{m}(r)-\tau_{m}(\gamma)= (r-\gamma)^{\prime}\nabla\tau_{m} (\gamma)+\frac{1}{2}(r-\gamma)^{\prime}\nabla^{2}\tau_{m}(\bar{r})(r-\gamma)\\ = & (r-\gamma)^{\prime}\nabla\tau_{m}(\gamma)+\frac{1}{2}(r-\gamma)^{\prime }\nabla^{2}\tau_{m}(\gamma)(r-\gamma)+\frac{1}{2}(r-\gamma)^{\prime}\left[ \nabla^{2}\tau_{m}(\bar{r})-\nabla^{2}\tau_{m}(\gamma)\right] (r-\gamma), \end{align*} and hence by Assumption C7 \begin{align} \mathbb{E}[\tau_{m}(r)]= & (r-\gamma)^{\prime}\mathbb{E}[\nabla\tau _{m}(\gamma)]+\frac{1}{2}(r-\gamma)^{\prime}\mathbb{E}[\nabla^{2}\tau _{m}(\gamma)](r-\gamma)+o(\Vert r-\gamma\Vert^{2})\nonumber\\ = & \frac{1}{2}(r-\gamma)^{\prime}\mathbb{V}_{\gamma}(r-\gamma)+o(\Vert r-\gamma\Vert^{2}), \end{align} where the second equality uses the fact that $\mathbb{E}[\nabla\tau_{m} (\gamma)]=0$ since $\mathbb{E}[\tau_{m}(r)]$ is maximized at $\gamma$. Then, applying ((ref)), ((ref)), and ((ref)) proves the lemma. \end{proof} \begin{proof} [Proof of Lemma (ref)]We can establish a representation for $\mathbb{E}[\mathcal{L}_{N}^{K}(r)|Z_{m}]$ using the same arguments as in proving Lemma (ref), but no longer integrating over $Z_{m}$. Specifically, a change of variables $u_{im}=v_{im}/\sigma_{N}$ gives \begin{align*} & \mathbb{E}[\mathcal{L}_{N}^{K}(r)|Z_{m}=z_{m}]=\int\sigma_{N}^{-2} K_{\sigma_{N},\gamma}(v_{im})B(v_{i},v_{m},w_{i},w_{m})S_{im}(r)\text{d} F_{V,W}(v_{i},w_{i})\\ = & \int K_{\gamma}(u_{im})B(v_{m}+u_{im}\sigma_{N},v_{m},w_{i},w_{m} )S_{im}(r)f_{V,W}(v_{m}+u_{im}\sigma_{N},w_{i})\text{d}w_{i}\text{d}u_{im}. \end{align*} The lead term of the $\kappa_{\gamma}^{\text{th}}$-order Taylor expansion inside the integral around $v_{m}$ is \begin{align*} & \int K_{\gamma}(u_{im})B(v_{m},v_{m},w_{i},w_{m})S_{im}(r)f_{V,W} (v_{m},w_{i})\text{d}w_{i}\text{d}u_{im}\\ = & \int B(v_{m},v_{m},w_{i},w_{m})S_{im}(r)f_{V,W}(v_{m},w_{i} )\text{d}w_{i}=\tau_{m}(r) \end{align*} by $\int K_{\gamma}(u)du=1$, and the sample average of the bias term is of order $o(\Vert r-\gamma\Vert^{2})$. Then, applying Lemma (ref) and Assumption C7, we can write \begin{align*} & \frac{2}{N}\sum_{m}\mathbb{E}[\mathcal{L}_{N}^{K}(r)|Z_{m}]-2\mathbb{E} [\mathcal{L}_{N}^{K}(r)]\\ = & \frac{2}{N}\sum_{m}\tau_{m}(r)-(r-\gamma)^{\prime}\mathbb{V}_{\gamma }(r-\gamma)+o_{p}(\Vert r-\gamma\Vert^{2})\\ = & \frac{2}{N}\sum_{m}(r-\gamma)^{\prime}\nabla\tau_{m}(\gamma)+\frac{1} {N}\sum_{m}(r-\gamma)^{\prime}\left( \nabla^{2}\tau_{m}(\gamma)-\mathbb{V} _{\gamma}\right) (r-\gamma)+o_{p}(\Vert r-\gamma\Vert^{2}). \end{align*} Then the desired result follows by noticing that $N^{-1}\sum_{m}\nabla^{2} \tau_{m}(\gamma)-\mathbb{V}_{\gamma}=o_{p}(1)$ by the SLLN. \end{proof} \begin{proof} [Proof of Lemma (ref)]By construction, we can write \[ \rho_{N}(r)=\frac{1}{N(N-1)}\sum_{i\neq m}\rho_{im}(r), \] where $\rho_{im}(r)\equiv(K_{\sigma_{N},\gamma}(V_{im})h_{im}(r)-2N^{-1} \sum_{i}\mathbb{E}[K_{\sigma_{N},\gamma}(V_{im})h_{im}(r)|Z_{i}]+\mathbb{E} [K_{\sigma_{N},\gamma}(V_{im})h_{im}(r)])/\sigma_{N}^{2}$. Note that $\rho_{im}(\gamma)=0$ and $|\rho_{im}(r)|$ is bounded by a multiple of $M/\sigma_{N}^{2}$ where $M$ is a positive constant. Define $\mathcal{F} _{N}^{\ast}=\{\rho_{im}^{\ast}(r)|r\in\mathcal{R}\}$ where $\rho_{im}^{\ast }(r)\equiv\sigma_{N}^{2}\rho_{im}(r)/M$. Then the Euclidean properties of the ($P$-degenerate) class of functions $\mathcal{F}_{N}^{\ast}$ are deduced using similar arguments for proving Lemma (ref) in combination with Corollary 17 and Corollary 21 in NolanPollard1987. As $\int K_{\sigma_{N},\gamma}(v)^{p}$d$v=O(\sigma_{N}^{2})$ for $p=1,2$, we have $\sup_{r\in\mathcal{R}_{N}}\mathbb{E}[\rho_{im}^{\ast}(r)^{2}]=O(\sigma _{N}^{2})$. Then applying Theorem 3 of Sherman1994ET gives \[ \frac{1}{N(N-1)}\sum_{i\neq m}\rho_{im}^{\ast}(r)=O_{p}(N^{-1}\sigma _{N}^{\alpha}), \] where $0<\alpha<1$ and hence $q_{N}(r)=O_{p}(N^{-1}\sigma_{N}^{\alpha -2})=O_{p}(N^{-1}\sigma_{N}^{-2})$. \end{proof} \begin{proof} [Proof of Lemma (ref)]The first step is to plug ((ref)) into ((ref)) so that $\Delta\mathcal{L}_{N}^{K}(r)$ can be expressed as \begin{equation} \frac{1}{\sigma_{N}^{3}N(N-1)(N-2)}\sum_{i\neq m\neq l}[\nabla_{1} K_{\sigma_{N},\gamma}(V_{im})X_{im1}^{\prime}\psi_{l,\beta}+\nabla _{2}K_{\sigma_{N},\gamma}(V_{im})X_{im2}^{\prime}\psi_{l,\beta}]h_{im}(r) \end{equation} plus a remainder term of higher order since $\sqrt{N}\sigma_{N}\rightarrow \infty$. ((ref)) is a third order U-process. We use the U-statistic decomposition found in Sherman1994ET (see also Serfling2009) to derive a linear representation. Note that ((ref)) has unconditional mean zero, so as is its mean conditional on each of its first two arguments $(Z_i,Z_m)$. Furthermore, by Theorem 3 of Sherman1994ET and similar arguments for proving Lemma (ref), the remainder term of the decomposition (projection) is of order $O_{p}(N^{-1}\sigma_{N}^{-2})$. Hence it suffices to derive a linear representation for its mean conditional on its third argument: \[ \frac{1}{\sigma_{N}^{3}N}\sum_{l}\int B(v_{i},v_{m},w_{i},w_{m})S_{im} (r)\nabla K_{\sigma_{N},\gamma}(v_{im})^{\prime}\left( \begin{array} [c]{c} x_{im1}^{\prime}\psi_{l,\beta}\\ x_{im2}^{\prime}\psi_{l,\beta} \end{array} \right) \text{d}F_{X,W}(x_{i},w_{i})\text{d}F_{X,W}(x_{m},w_{m}). \] where $F_{X,W}(\cdot,\cdot)$ denotes the joint distribution function of $(X_{i},W_{i})$. While the above integral is expressed w.r.t. $(x_{i},w_{i})$ and $(x_{m},w_{m})$, it will prove convenient to express the integral in terms of $v_{i}$ and $v_{m}$. We do so as follows: \begin{equation} \frac{1}{\sigma_{N}^{3}N}\sum_{l}\left( \int\nabla K_{\sigma_{N},\gamma }(v_{im})^{\prime}G(v_{i},v_{m},r)f_{V}(v_{i})f_{V}(v_{m})\text{d} v_{i}\text{d}v_{m}\right) \psi_{l,\beta}. \end{equation} Apply a change of variables in ((ref)) with $u_{im}=v_{im}/\sigma_{N}$ to obtain \begin{align*} & \frac{1}{\sigma_{N}^{3}N}\sum_{l}\left( \int\nabla K_{\sigma_{N},\gamma }(v_{im})^{\prime}G(v_{i},v_{m},r)f_{V}(v_{i})f_{V}(v_{m})\text{d} v_{i}\text{d}v_{m}\right) \psi_{l,\beta}\\ = & \frac{1}{\sigma_{N}N}\sum_{l}\left( \int\nabla K_{\gamma} (u_{im})^{\prime}G(v_{m}+u_{im}\sigma_{N},v_{m},r)f_{V}(v_{m}+u_{im}\sigma _{N})f_{V}(v_{m})\text{d}u_{im}\text{d}v_{m}\right) \psi_{l,\beta}. \end{align*} As $\int\nabla K_{\gamma}(u)^{\prime}$d$u=0$ and $\int u_{j}\nabla _{j}K_{\gamma}(u)$d$u=-1$ for $j=1,2$, a second-order expansion inside the integral around $u_{im}\sigma_{N}=0$ yields the lead term\footnote{The second-order term is of order $O_{p}(\delta_{N}\sigma_{N}/\sqrt{N})$, which will be $o_{p}(\Vert r-\gamma\Vert^{2})$.} \[ -\frac{1}{N}\sum_{l}\left( \int\nabla_{1}\mu(v_{m},v_{m},r)f_{V} (v_{m})\text{d}v_{m}\right) \psi_{l,\beta}. \] We next apply a second-order Taylor expansion of $\nabla_{1}\mu(v_{m} ,v_{m},r)$ around $\gamma$ to yield \[ (r-\gamma)^{\prime}\frac{1}{N}\sum_{l}\left( -\int\nabla_{13}^{2}\mu (v_{m},v_{m},\gamma)f_{V}(v_{m})\text{d}v_{m}\right) \psi_{l,\beta} +o_{p}(\Vert r-\gamma\Vert^{2}), \] which concludes the proof of this lemma. \end{proof} \begin{proof} [Proof of Lemma (ref)] Recall that $ V_{im}\left( \beta \right) =\left( X_{im1}^{\prime }\beta ,X_{im2}^{\prime }\beta \right) .$ The following calculation is useful:\textbf{\ } \begin{eqnarray} &&K_{\sigma _{N},\gamma } ( V_{im} ( \hat{\beta} ) ) -K_{\sigma _{N},\gamma }\left( V_{im}\left( \beta \right) \right) \notag\\ &=&K\left( \frac{X_{im1}^{\prime }\hat{\beta}}{\sigma _{N}}\right) K\left( \frac{X_{im2}^{\prime }\hat{\beta}}{\sigma _{N}}\right) -K\left( \frac{ X_{im1}^{\prime }\beta }{\sigma _{N}}\right) K\left( \frac{X_{im2}^{\prime }\beta }{\sigma _{N}}\right) \notag \\ &=&K^{\prime }\left( \frac{X_{im1}^{\prime }\beta }{\sigma _{N}}\right) K\left( \frac{X_{im2}^{\prime }\beta }{\sigma _{N}}\right) \frac{ X_{im1}^{\prime } ( \hat{\beta}-\beta ) }{\sigma _{N}}\notag\\ &&+K\left( \frac{X_{im1}^{\prime }\beta }{\sigma _{N}}\right) K^{\prime }\left( \frac{ X_{im2}^{\prime }\beta }{\sigma _{N}}\right) \frac{X_{im2}^{\prime } ( \hat{\beta}-\beta ) }{\sigma _{N}}+O_{p} ( N^{-1}\sigma _{N}^{-2} ) , \end{eqnarray} where the last line applies the Taylor expansion. Suppose we have a random vector $q_{im}$ such that $\mathbb{E}[q_{im}|V_{im}\left( \beta \right) =v] $ exists and is $\kappa _{\gamma }$-th differentiable in $v,$ with derivatives up to $\kappa _{\gamma }$-th being uniformly bounded. We make the following claim \begin{equation} \frac{1}{N\left( N-1\right) }\sum_{i\neq m}K^{\prime }\left( \frac{X_{im1}^{\prime }\beta }{\sigma _{N}}\right) K\left( \frac{ X_{im2}^{\prime }\beta }{\sigma _{N}}\right) q_{im}=O_{p}( N^{-1/2}\sigma _{N}^{2}), \end{equation} whose proof is deferred to the end. Using ((ref)), we can write \begin{eqnarray} &&\frac{1}{\sigma _{N}^{2}N\left( N-1\right) }\sum_{i\neq m}K^{\prime }\left( \frac{X_{im1}^{\prime }\beta }{\sigma _{N}}\right) K\left( \frac{ X_{im2}^{\prime }\beta }{\sigma _{N}}\right) \frac{X_{im1}^{\prime }( \hat{\beta}-\beta ) }{\sigma _{N}}Y_{im\left( 1,1\right) }\text{sgn} \left( W_{im}^{\prime }\gamma \right) \notag \\ &=&\left[ \frac{1}{\sigma _{N}^{2}N\left( N-1\right) }\sum_{i\neq m}K^{\prime }\left( \frac{X_{im1}^{\prime }\beta }{\sigma _{N}}\right) K\left( \frac{X_{im2}^{\prime }\beta }{\sigma _{N}}\right) X_{im1}^{\prime }Y_{im\left( 1,1\right) }\text{sgn}\left( W_{im}^{\prime }\gamma \right) \right] \frac{( \hat{\beta}-\beta) }{\sigma _{N}} \notag \\ &=&O_{p}(N^{-1/2}( \hat{\beta}-\beta)\sigma _{N}^{-1} ) =O_{p}( N^{-1}\sigma _{N}^{-1}) , \end{eqnarray} where we let $q_{im}=X_{im1}^{\prime }Y_{im\left( 1,1\right) }$sgn$\left( W_{im}^{\prime }\gamma \right)$ and use the result $\hat{\beta}-\beta =O_{p} ( N^{-1/2} ) .$ Clearly, this also holds if we replace $ X_{im1}^{\prime } ( \hat{\beta}-\beta ) $ with $X_{im2}^{\prime } ( \hat{\beta}-\beta ) .$ Combine ((ref)) and ((ref)) to obtain \begin{eqnarray*} &&\frac{1}{\sigma _{N}^{2}N\left( N-1\right) }\sum_{i\neq m}\left[ K_{\sigma _{N},\gamma } ( V_{im} ( \hat{\beta} ) ) -K_{\sigma _{N},\gamma }\left( V_{im}\left( \beta \right) \right) \right] Y_{im\left( 1,1\right) }\text{sgn}\left( W_{im}^{\prime }\gamma \right) \\ &=&2\cdot O_{p}\left( N^{-1}\sigma _{N}^{-1}\right) +O_{p}\left( N^{-1}\sigma _{N}^{-2}\right) =O_{p}\left( N^{-1}\sigma _{N}^{-2}\right) , \end{eqnarray*} as desired. The remaining task is to show the claim in ((ref)). Some standard results on U-statistics imply that the variance of ((ref)) is $O_{p}( N^{-1})$. So we only need show the expectation of the U-statistic is $O( N^{-1/2})$. Under Assumption C5, we can calculate the expectation of the summand of ((ref)) as follows: \begin{eqnarray*} &&\mathbb{E}\left[ K^{\prime }\left( \frac{X_{im1}^{\prime }\beta }{\sigma _{N}}\right) K\left( \frac{X_{im2}^{\prime }\beta }{\sigma _{N}}\right) q_{im}\right] =\mathbb{E}\left[ K^{\prime }\left( \frac{X_{im1}^{\prime }\beta }{\sigma _{N}}\right) K\left( \frac{X_{im2}^{\prime }\beta }{\sigma _{N}}\right) \mathbb{E}\left( q_{im}|V_{im}(\beta) \right) \right] \\ &=&\int \int K^{\prime }\left( \frac{v_{1}}{\sigma _{N}}\right) K\left( \frac{v_{2}}{\sigma _{N}}\right) \mathbb{E}[q_{im}|V_{im}(\beta)=(v_1,v_2)] f_{V_{im}(\beta)}( v_{1},v_{2}) dv_{1}dv_{2}\\ &=&\sigma_N^2\int \int K^{\prime }\left( u_{1}\right) K\left( u_{2}\right) \mathbb{E} [q_{im}|V_{im}(\beta)=(u_{1}\sigma _{N},u_{2}\sigma _{N})]f_{V_{im}(\beta)}( u_{1}\sigma _{N},u_{2}\sigma _{N}) du_{1}du_{2} \\ &=&\sigma_N^2\mathbb{E} [q_{im}|V_{im}(\beta)=(0,0)]f_{V_{im}(\beta)}(0, 0) \int \int K^{\prime }\left( u_{1}\right) K\left( u_{2}\right) du_{1}du_{2}+O_{p} ( \sigma_N^4 ) \\ &=&0+O_{p} ( N^{-1/2} ) =O_{p} ( N^{-1/2} ), \end{eqnarray*} where the last line holds by Assumptions C6 and C7. This shows the claim. \end{proof} \begin{proof} [Proof of Lemma (ref)] Applying similar arguments for obtaining ((ref)), we can write \begin{eqnarray*} \mathcal{\hat{L}}_{N}^{K}\left( r\right) &=&\frac{1}{2}\left( r-\gamma \right) ^{\prime }\mathbb{V}_{\gamma }\left( r-\gamma \right) +O_{p} ( N^{-1/2}\left\Vert r-\gamma \right\Vert ) +o_{p} ( \left\Vert r-\gamma \right\Vert ^{2} ) \\ &&+\frac{1}{\sigma _{N}^{2}N\left( N-1\right) }\sum_{i\neq m}K_{\sigma _{N},\gamma } ( V_{im} ( \hat{\beta} ) ) Y_{im\left( 1,1\right) }\text{sgn}\left( W_{im}^{\prime }\gamma \right) +O_{p}\left( N^{-1}\sigma _{N}^{-2}\right) . \end{eqnarray*} Then, using the result in Lemma (ref), we have \begin{eqnarray*} \mathcal{\hat{L}}_{N}^{K}\left( r\right) &=&\frac{1}{2}\left( r-\gamma \right) ^{\prime }\mathbb{V}_{\gamma }\left( r-\gamma \right) +O_{p} ( N^{-1/2}\left\Vert r-\gamma \right\Vert ) +o_{p} ( \left\Vert r-\gamma \right\Vert ^{2} ) \\ &&+\frac{1}{\sigma _{N}^{2}N\left( N-1\right) }\sum_{i\neq m}K_{\sigma _{N},\gamma }\left( V_{im}\left( \beta \right) \right) Y_{im\left( 1,1\right) }\text{sgn}\left( W_{im}^{\prime }\gamma \right) +o_{p} ( N^{-1/2} ) +O_{p} ( N^{-1}\sigma _{N}^{-2} ) . \end{eqnarray*} Since $\ N^{1/2}\sigma _{N}^{2}\rightarrow \infty $ by Assumption C7, $O_{p}\left( N^{-1}\sigma _{N}^{-2}\right) =o_{p}\left( N^{-1/2}\right)$, which completes the proof. \end{proof} \begin{proof} [Proof of Lemma (ref)] Using the definition that $ V_{im}\left( \beta \right) =\left( X_{im1}^{\prime }\beta ,X_{im2}^{\prime }\beta \right) $, we can write \begin{eqnarray*} &&\mathbb{E}\left[ \sigma _{N}^{-2}K_{\sigma _{N},\gamma }\left( V_{im}\left( \beta \right) \right) Y_{im\left( 1,1\right) }\text{sgn}\left( W_{im}^{\prime }\gamma \right) |Z_{i}\right] \\ &=&\mathbb{E}\left[ \left. \sigma _{N}^{-2}K\left( \frac{X_{im1}^{\prime }\beta }{\sigma _{N}}\right) K\left( \frac{X_{im2}^{\prime }\beta }{\sigma _{N}}\right) Y_{im\left( 1,1\right) }\text{sgn}\left( W_{im}^{\prime }\gamma \right) \right\vert Z_{i}\right] \\ &=&\mathbb{E}\left[ \left. \sigma _{N}^{-2}K\left( \frac{X_{im1}^{\prime }\beta }{\sigma _{N}}\right) K\left( \frac{X_{im2}^{\prime }\beta }{\sigma _{N}}\right) \mathbb{E}\left[ Y_{im\left( 1,1\right) }\text{sgn}\left( W_{im}^{\prime }\gamma \right) |Z_{i},X_{m1},X_{m2}\right] \right\vert Z_{i} \right] \\ &=&f_{V_{im}(\beta)}(0)\mathbb{E}\left[ Y_{im\left( 1,1\right) }\text{sgn}\left( W_{im}^{\prime }\gamma \right) |Z_{i},V_{im}\left( \beta \right) =0 \right] +o_{p} ( N^{-1/2} ), \end{eqnarray*} where the last line uses the same argument as in proving Lemma (ref). \end{proof} \begin{proof} [Proof of Lemma (ref)]We verify the conditions in Assumption M, specifically, M.i, M.ii, and M.iii, in SeoOtsu2018 one by one. Recall that \begin{align*} \phi_{Ni}\left( b\right) & \equiv\sum_{t>s}\sum_{d\in\mathcal{D} }\{\mathcal{K}_{h_{N}}(X_{i2ts},W_{its})Y_{idst}\left( -1\right) ^{d_{1} }\left( 1[X_{i1ts}^{\prime}b>0]-1[X_{i1ts}^{\prime}\beta>0]\right) \\ & +\mathcal{K}_{h_{N}}(X_{i1ts},W_{its})Y_{idst}\left( -1\right) ^{d_{2} }\left( 1[X_{i2st}^{\prime}b>0]-1[X_{i2st}^{\prime}\beta>0]\right) \}, \end{align*} The \textquotedblleft$h_{n}$\textquotedblright\ in SeoOtsu2018 needs to be set as $h_{N}^{k_{1}+k_{2}}$ for the term $\phi_{Ni}\left( b\right) $. Similarly, the \textquotedblleft$h_{n}$\textquotedblright\ in SeoOtsu2018 needs to be set as $\sigma_{N}^{2k_{1}}$ for the term $\varphi_{Ni}\left( r\right) $. We conduct the analysis for $\phi _{Ni}\left( b\right) $ first$,$ and the results for $\varphi_{Ni}\left( r\right) $ follow similarly. For $\phi_{Ni}\left( b\right) ,$ we analyze the term \[ \mathcal{K}_{h_{N}}(X_{i2ts},W_{its})Y_{idst}\left( -1\right) ^{d_{1} }\left( 1[X_{i1ts}^{\prime}b>0]-1[X_{i1ts}^{\prime}\beta>0]\right) , \] and the remaining terms can be analyzed similarly, thanks to the similar structure of those terms. \textbf{On Assumption M.i in} SeoOtsu2018. Note that \begin{align} & \mathbb{E}[\mathcal{K}_{h_{N}}(X_{i2ts},W_{its})Y_{idst}\left( -1\right) ^{d_{1}}\left( 1\left[ X_{i1ts}^{\prime}b>0\right] -1\left[ X_{i1ts} ^{\prime}\beta>0\right] \right) ]\\ = & \int_{\mathbb{R}^{k_{2}}}\int_{\mathbb{R}^{k_{1}}}\mathbb{E} [Y_{idst}\left( -1\right) ^{d_{1}}\left( 1\left[ X_{i1ts}^{\prime }b>0\right] -1\left[ X_{i1ts}^{\prime}\beta>0\right] \right) |X_{i2ts}=x_{2},W_{its}=w]\nonumber\\ & \cdot\mathcal{K}_{h_{N}}(x_{2},w)f_{X_{2ts},W_{ts}}\left( x_{2},w\right) \text{d}x_{2}\text{d}w\nonumber\\ = & \mathbb{E}[Y_{idst}\left( -1\right) ^{d_{1}}\left( 1\left[ X_{i1ts}^{\prime}b>0\right] -1\left[ X_{i1ts}^{\prime}\beta>0\right] \right) |X_{i2ts}=0,W_{its}=0]f_{X_{2ts},W_{ts}}\left( 0,0\right) +O(h_{N}^{2}),\nonumber \end{align} where the second equality holds by change of variables and Taylor expansion, and the small order term holds by the properties of the kernel function and the boundedness conditions imposed in Assumptions P5--P7. Further, by Assumption P8, the small order term satisfies $O(h_{N}^{2})=o((Nh_{N} ^{k_{1}+k_{2}})^{-2/3}).$ As a result, we only need to focus on the lead term \begin{align} & \mathbb{E}[Y_{idst}\left( -1\right) ^{d_{1}}\left( 1\left[ X_{i1ts}^{\prime}b>0\right] -1\left[ X_{i1ts}^{\prime}\beta>0\right] \right) |X_{i2ts}=0,W_{its}=0]\nonumber\\ = & \mathbb{E}[\mathbb{E}[Y_{idst}\left( -1\right) ^{d_{1}}|X_{i1ts} ,X_{i2ts}=0,W_{its}=0]\left( 1\left[ X_{i1ts}^{\prime}b>0\right] -1\left[ X_{i1ts}^{\prime}\beta>0\right] \right) |X_{i2ts}=0,W_{its}=0]\nonumber\\ \equiv & \mathbb{E}[\kappa_{dts}^{\left( 1\right) }\left( X_{i1ts}\right) \left( 1\left[ X_{i1ts}^{\prime}b>0\right] -1\left[ X_{i1ts}^{\prime} \beta>0\right] \right) |X_{i2ts}=0,W_{its}=0]\nonumber\\ = & \int_{\mathbb{R}^{k_{1}}}\kappa_{dts}^{\left( 1\right) }\left( x\right) \left( 1\left[ x^{\prime}b>0\right] -1\left[ x^{\prime} \beta>0\right] \right) f_{X_{1ts}|\left\{ X_{2ts}=0,W_{ts}=0\right\} }\left( x\right) \text{d}x\nonumber\\ \equiv & \int_{\mathbb{R}^{k_{1}}}\kappa_{dts}^{\left( 1\right) }\left( x\right) \left( 1\left[ x^{\prime}b>0\right] -1\left[ x^{\prime} \beta>0\right] \right) f_{X_{1ts}|\left\{ X_{2ts}=0,W_{ts}=0\right\} }\left( x\right) \text{d}x, \end{align} where to ease notations we denote \[ \kappa_{dts}^{\left( 1\right) }\left( x\right) =\mathbb{E}[Y_{idst}\left( -1\right) ^{d_{1}}|X_{i1ts}=x,X_{i2ts}=0,W_{its}=0] \] in the third line. We now get the first and second derivatives of the above term w.r.t. $b$ around $\beta.$ Since the calculation is the classical differential geometry and very similar to that in Sections 5 and 6.4 of KimPollard1990 and Section B.1 of SeoOtsu2018, we only present key steps and omit some standard similar details. First, define the mapping \[ T_{b}= ( I-\left\Vert b\right\Vert _{2}^{-2}bb^{\prime} ) ( I-\beta \beta^{\prime} ) +\left\Vert b\right\Vert _{2}^{-2}b\beta^{\prime}. \] Then $T_{b}$ maps $\left\{ X_{i1ts}^{\prime}b>0\right\} $ onto $\left\{ X_{i1ts}^{\prime}\beta>0\right\} $ and $\left\{ X_{i1ts}^{\prime }b=0\right\} $ onto $\left\{ X_{i1ts}^{\prime}\beta=0\right\} .$ With equation ((ref)), equations (5.2) and (5.3) in KimPollard1990 imply \begin{align*} & \frac{\partial}{\partial b}\mathbb{E}\left[ \left. Y_{idst}\left( -1\right) ^{d_{1}}\left( 1\left[ X_{i1ts}^{\prime}b>0\right] -1\left[ X_{i1ts}^{\prime}\beta>0\right] \right) \right\vert X_{i2ts}=0,W_{its} =0\right] \\ & =\frac{\partial}{\partial b}\int_{\mathbb{R}^{k_{1}}}\kappa_{dts}^{\left( 1\right) }\left( x\right) \left( 1\left[ x^{\prime}b>0\right] -1\left[ x^{\prime}\beta>0\right] \right) f_{X_{1ts}|\left\{ X_{2ts}=0,W_{ts} =0\right\} }\left( x\right) \text{d}x,\\ & =\left\Vert b\right\Vert _{2}^{-2}b^{\prime}\beta\left( I-\left\Vert b\right\Vert _{2}^{-2}bb^{\prime}\right) \int1\left[ x^{\prime} \beta=0\right] \kappa_{dts}^{\left( 1\right) }\left( T_{b}x\right) xf_{X_{1ts}|\left\{ X_{2ts}=0,W_{ts}=0\right\} }\left( T_{b}x\right) \text{d}\sigma_{0}^{\left( 1\right) }, \end{align*} where $\sigma_{0}^{\left( 1\right) }$ is the surface measure of $\left\{ X_{i1ts}^{\prime}\beta=0\right\} .$ Note that $T_{\beta}x=x,$ then \[ 1\left[ x^{\prime}\beta=0\right] \kappa_{dts}^{\left( 1\right) }\left( T_{\beta}x\right) =1\left[ x^{\prime}\beta=0\right] \kappa_{dts}^{\left( 1\right) }\left( x\right) =0 \] because \begin{align*} \left. \kappa_{dts}^{\left( 1\right) }\left( X_{i1t}\right) \right\vert _{x^{\prime}\beta=0} & =\mathbb{E}\left[ \left. Y_{idst}\left( -1\right) ^{d_{1}}\right\vert X_{i1ts}^{\prime}\beta=0,X_{i2ts}=0,W_{its}=0\right] \\ & =\mathbb{E}\left[ \left. \left( Y_{idt}-Y_{ids}\right) \left( -1\right) ^{d_{1}}\right\vert X_{i1t}^{\prime}\beta=X_{i1s}^{\prime} \beta,X_{i2t}=X_{i2s},W_{it}=W_{is}\right] \\ & =0, \end{align*} where the last line holds by Assumption P2 that $\xi_{s}\overset{d}{=}\xi_{t}| ( \alpha,Z^{T} ) .$ This implies that \[ \left. \frac{\partial}{\partial b}\mathbb{E}\left[ \left. Y_{idst}\left( -1\right) ^{d_{1}}\left( 1\left[ X_{i1ts}^{\prime}b>0\right] -1\left[ X_{i1ts}^{\prime}\beta>0\right] \right) \right\vert X_{i2ts}=0,W_{its} =0\right] \right\vert _{b=\beta}=0. \] For the same reason, the nonzero component of the second order derivative at $b=\beta$ only comes from the derivative of $\kappa_{dts}^{\left( 1\right) }\left( X_{i1ts}\right) .$ Notice that $\left. \frac{\partial\kappa _{dts}^{\left( 1\right) }\left( T_{b}x\right) }{\partial b}\right\vert _{b=\beta}=-\left( \frac{\partial\kappa_{dts}^{\left( 1\right) }\left( x\right) }{\partial x}^{\prime}\beta\right) x,$ we have \begin{align*} & \left. \frac{\partial^{2}}{\partial b\partial b^{\prime}}\mathbb{E}\left[ \left. Y_{idst}\left( -1\right) ^{d_{1}}\left( 1\left[ X_{i1ts}^{\prime }b>0\right] -1\left[ X_{i1ts}^{\prime}\beta>0\right] \right) \right\vert X_{i2ts}=0,W_{its}=0\right] \right\vert _{b=\beta}\\ = & -\int1\left[ x^{\prime}\beta=0\right] \left( \frac{\partial \kappa_{dts}^{\left( 1\right) }\left( x\right) }{\partial x}^{\prime} \beta\right) f_{X_{1ts}|\left\{ X_{2ts}=0,W_{ts}=0\right\} }\left( x\right) xx^{\prime}\text{d}\sigma_{0}^{\left( 1\right) } \equiv \mathbb{V}_{dts}^{\left( 1\right) }. \end{align*} With the first and second derivatives obtained, we can move on to apply the Taylor expansion to the expectation term in equation ((ref)) around $b=\beta$\ as \begin{align*} & \mathbb{E}\left[ \left. Y_{idst}\left( -1\right) ^{d_{1}}\left( 1\left[ X_{i1ts}^{\prime}b>0\right] -1\left[ X_{i1ts}^{\prime} \beta>0\right] \right) \right\vert X_{i2ts}=0,W_{its}=0\right] \\ = & \frac{1}{2}\left( b-\beta\right) ^{\prime}\mathbb{V}_{dts}^{\left( 1\right) }\left( b-\beta\right) +o\left( \left\Vert b-\beta\right\Vert ^{2}\right) . \end{align*} Substitute this back to equation ((ref)), we have \begin{align*} & \mathbb{E}\left[ \mathcal{K}_{h_{N}}(X_{i2ts},W_{its})Y_{idst}\left( -1\right) ^{d_{1}}\left( 1\left[ X_{i1ts}^{\prime}b>0\right] -1\left[ X_{i1ts}^{\prime}\beta>0\right] \right) \right] \\ = & \frac{1}{2}\left( b-\beta\right) ^{\prime}\mathbb{V}_{dts}^{\left( 1\right) }\left( b-\beta\right) +o\left( \left\Vert b-\beta\right\Vert ^{2}\right) +o\left( \left( Nh_{N}^{k_{1}+k_{2}}\right) ^{-2/3}\right) . \end{align*} We can similarly define \[ \mathbb{V}_{dts}^{\left( 2\right) }\equiv-\int1\left[ x^{\prime} \beta=0\right] \left( \frac{\partial\kappa_{dts}^{\left( 2\right) }\left( x\right) }{\partial x}^{\prime}\beta\right) f_{X_{2ts}|\left\{ X_{1ts}=0,W_{ts}=0\right\} }\left( x\right) xx^{\prime}\text{d}\sigma _{0}^{\left( 2\right) }, \] where \[ \kappa_{dts}^{\left( 2\right) }\left( x\right) \equiv\mathbb{E}\left[ \left. Y_{idst}\left( -1\right) ^{d_{2}}\right\vert X_{i2ts}=x,X_{i1ts} =0,W_{its}=0\right] , \] and $\sigma_{0}^{\left( 2\right) }$ is the surface measure of $\left\{ X_{i2ts}^{\prime}\beta=0\right\} $. A similar derivation yields \begin{align*} & \mathbb{E}\left[ \mathcal{K}_{h_{N}}(X_{i1ts},W_{its})Y_{idst}\left( -1\right) ^{d_{2}}\left( 1\left[ X_{i2st}^{\prime}b>0\right] -1\left[ X_{i2st}^{\prime}\beta>0\right] \right) \right] \\ = & \frac{1}{2}\left( b-\beta\right) ^{\prime}\mathbb{V}_{dts}^{\left( 2\right) }\left( b-\beta\right) +o\left( \left\Vert b-\beta\right\Vert ^{2}\right) +o\left( \left( Nh_{N}^{k_{1}+k_{2}}\right) ^{-2/3}\right) . \end{align*} Put everything so far together, we have \begin{equation} \mathbb{E}\left[ \phi_{Ni}\left( b\right) \right] =\frac{1}{2}\left( b-\beta\right) ^{\prime}\left[ \sum_{t>s}\sum_{d\in\mathcal{D}}\left( \mathbb{V}_{dts}^{\left( 1\right) }+\mathbb{V}_{dts}^{\left( 2\right) }\right) \right] \left( b-\beta\right) +o ( \left\Vert b-\beta\right\Vert ^{2} ) +o ( ( Nh_{N}^{k_{1}+k_{2}} ) ^{-2/3}) . \end{equation} Assumption M.i in SeoOtsu2018 is then verified by \begin{align*} \mathbb{V} \equiv & \sum_{t>s}\sum_{d\in\mathcal{D}}\left( \mathbb{V} _{dts}^{\left( 1\right) }+\mathbb{V}_{dts}^{\left( 2\right) }\right) .\\ = & -\sum_{t>s}\sum_{d\in\mathcal{D}}\left\{ \int1\left[ x^{\prime} \beta=0\right] \left( \frac{\partial\kappa_{dts}^{\left( 1\right) }\left( x\right) }{\partial x}^{\prime}\beta\right) f_{X_{1ts}|\left\{ X_{2ts}=0,W_{ts}=0\right\} }\left( x\right) xx^{\prime}\text{d}\sigma _{0}^{\left( 1\right) }\right. \\ & \left. +\int1\left[ x^{\prime}\beta=0\right] \left( \frac {\partial\kappa_{dts}^{\left( 2\right) }\left( x\right) }{\partial x}^{\prime}\beta\right) f_{X_{2ts}|\left\{ X_{1ts}=0,W_{ts}=0\right\} }\left( x\right) xx^{\prime}\text{d}\sigma_{0}^{\left( 2\right) }\right\} . \end{align*} We can obtain similar results for $\varphi_{Ni}\left( r\right) $ using exactly the same line of analysis, i.e., \begin{equation} \mathbb{E}\left[ \varphi_{Ni}\left( r\right) \right] =\frac{1}{2}\left( r-\gamma\right) ^{\prime}\mathbb{W}\left( r-\gamma\right) +o ( \left\Vert r-\gamma\right\Vert ^{2} ) +o( ( N\sigma_{N}^{2k_{1}} ) ^{-2/3}) , \end{equation} where \[ \mathbb{W\equiv}-\sum_{t>s}\int1\left[ w^{\prime}\gamma=0\right] \left( \frac{\partial\kappa_{\left( 1,1\right) ts}^{\left( 3\right) }\left( w\right) }{\partial w}^{\prime}\gamma\right) f_{W_{ts}|\left\{ X_{1ts}=0,X_{2ts}=0\right\} }\left( w\right) ww^{\prime}\text{d}\sigma_{0} \] with \[ \kappa_{\left( 1,1\right) ts}^{\left( 3\right) }\left( w\right) \equiv\mathbb{E}\left[ Y_{i(1,1)ts}|W_{its}=w,X_{i1ts}=0,X_{i2ts}=0\right] \] and $\sigma_{0}^{\left( 3\right) }$ being the surface measure of $\left\{ W_{its}^{\prime}\gamma=0\right\} $. \textbf{On Assumption M.ii in }SeoOtsu2018\textbf{. }Again, we first verify this condition for \[ \mathcal{K}_{h_{N}}(X_{i2ts},W_{its})Y_{idst}\left( -1\right) ^{d_{1} }\left( 1\left[ X_{i1ts}^{\prime}b>0\right] -1\left[ X_{i1ts}^{\prime }\beta>0\right] \right) , \] and other components in $\phi_{Ni}\left( b\right) $ follow similarly. We evaluate norm of the difference of the above term at $b=b_{1}$ and $b=b_{2}$ multiplied by $h_{N}^{\left( k_{1}+k_{2}\right) /2}$: \begin{align} & h_{N}^{\left( k_{1}+k_{2}\right) /2}\left\Vert \mathcal{K}_{h_{N} }(X_{i2ts},W_{its})Y_{idst}\left( -1\right) ^{d_{1}}\left( 1\left[ X_{i1ts}^{\prime}b_{1}>0\right] -1\left[ X_{i1ts}^{\prime}b_{2}>0\right] \right) \right\Vert \nonumber\\ = & \left\{ \mathbb{E}\left[ \mathbb{E}\left[ \left. h_{N}^{k_{1}+k_{2} }\mathcal{K}_{h_{N}}^{2}(X_{i2ts},W_{its})Y_{idst}^{2}\right\vert X_{i1ts}\right] \left( 1\left[ X_{i1ts}^{\prime}b_{1}>0\right] -1\left[ X_{i1ts}^{\prime}b_{2}>0\right] \right) ^{2}\right] \right\} ^{1/2}\nonumber\\ = & \left\{ \mathbb{E}\left[ \mathbb{E}\left[ \left. h_{N}^{k_{1}+k_{2} }\mathcal{K}_{h_{N}}^{2}(X_{i2ts},W_{its})Y_{idst}^{2}\right\vert X_{i1ts}\right] \left\vert 1\left[ X_{i1ts}^{\prime}b_{1}>0\right] -1\left[ X_{i1ts}^{\prime}b_{2}>0\right] \right\vert \right] \right\} ^{1/2}\\ \geq & C_{1}\mathbb{E}\left[ \left\vert 1\left[ X_{i1ts}^{\prime} b_{1}>0\right] -1\left[ X_{i1ts}^{\prime}b_{2}>0\right] \right\vert \right] \geq C_{2}\left\Vert b-\beta\right\Vert _{2},\nonumber \end{align} where the second equality holds by the fact that the difference of two indicator functions can only take values $-1,0,$ or 1. $C_{1}$ and $C_{2}$ are two positive constants. Applying the same analysis to all other terms in $\phi_{Ni}\left( b\right) ,$ we have \[ h_{N}^{\left( k_{1}+k_{2}\right) /2}\left\Vert \phi_{Ni}\left( b_{1}\right) -\phi_{Ni}\left( b_{2}\right) \right\Vert \geq C_{3}\left\Vert b_{1}-b_{2}\right\Vert \] for some positive constant $C_{3}.$ Applying similar analysis to $\varphi_{Ni}\left( r\right) $ leads to \[ \sigma_{N}^{k_{1}}\left\Vert \varphi_{Ni}\left( r_{1}\right) -\varphi _{Ni}\left( r_{2}\right) \right\Vert \geq C_{4}\left\Vert r_{1} -r_{2}\right\Vert \] for some positive constant $C_{4}.$ \textbf{On Assumption M.iii in} SeoOtsu2018\textbf{. }We still begin with the analysis on \[ \mathcal{K}_{h_{N}}(X_{i2ts},W_{its})Y_{idst}\left( -1\right) ^{d_{1} }\left( 1\left[ X_{i1ts}^{\prime}b>0\right] -1\left[ X_{i1ts}^{\prime }\beta>0\right] \right) , \] and other components in $\phi_{Ni}\left( b\right) $ follow similarly. We evaluate the square of difference of the above term at $b=b_{1}$ and $b=b_{2},$ such that $\left\Vert b_{1}-b_{2}\right\Vert <\varepsilon$, multiplied by $h_{N}^{k_{1}+k_{2}}$: \begin{align*} & h_{N}^{k_{1}+k_{2}}\mathbb{E}\left[ \sup_{b_{1},b_{2}\in\mathcal{B} ,\left\Vert b_{1}-b_{2}\right\Vert <\varepsilon}\left[ \mathcal{K}_{h_{N} }(X_{i2ts},W_{its})Y_{idst}\left( -1\right) ^{d_{1}}\left( 1\left[ X_{i1ts}^{\prime}b_{1}>0\right] -1\left[ X_{i1ts}^{\prime}b_{2}>0\right] \right) \right] ^{2}\right] \\ \leq & \mathbb{E}\left[ \mathbb{E}\left[ \left. h_{N}^{k_{1}+k_{2} }\mathcal{K}_{h_{N}}^{2}(X_{i2ts},W_{its})Y_{idst}^{2}\right\vert X_{i1ts}\right] \sup_{b_{1},b_{2}\in\mathcal{B},\left\Vert b_{1} -b_{2}\right\Vert <\varepsilon}\left\vert 1\left[ X_{i1ts}^{\prime} b_{1}>0\right] -1\left[ X_{i1ts}^{\prime}b_{2}>0\right] \right\vert \right] \\ \leq & C_{5}\mathbb{E}\left[ \sup_{b_{1},b_{2}\in\mathcal{B},\left\Vert b_{1}-b_{2}\right\Vert <\varepsilon}\left\vert 1\left[ X_{i1ts}^{\prime} b_{1}>0\right] -1\left[ X_{i1ts}^{\prime}b_{2}>0\right] \right\vert \right] \leq C_{6}\varepsilon, \end{align*} where the second line follows by the fact that the maximum of absolute value of the difference of two indicator functions is 1, and the last line holds by Assumptions P4 and P5. Apply the same analysis to all other terms in $\phi_{Ni}\left( b\right) $ to obtain \[ h_{N}^{k_{1}+k_{2}}\mathbb{E}\left[ \sup_{b_{1},b_{2}\in\mathcal{B} ,\left\Vert b_{1}-b_{2}\right\Vert <\varepsilon}\left[ \phi_{Ni}\left( b_{1}\right) -\phi_{Ni}\left( b_{2}\right) \right] ^{2}\right] \leq C_{7}\varepsilon \] for some positive constant $C_{7}.$ Similar analysis on $\varphi_{Ni}\left( r\right) $ leads to \[ h_{N}^{2k_{1}}\mathbb{E}\left[ \sup_{r_{1},r_{2}\in\mathcal{R},\left\Vert r_{1}-r_{2}\right\Vert <\varepsilon}\left[ \varphi_{Ni}\left( r_{1}\right) -\varphi_{Ni}\left( r_{2}\right) \right] ^{2}\right] \leq C_{8}\varepsilon \] for some positive constant $C_{8}.$ \end{proof} \begin{proof} [Proof of Lemma (ref)]The first two equalities are direct results from Lemma B.1 by Taylor expansions. Taking $b=\beta+\rho( Nh_{N} ^{k_{1}+k_{2}} ) ^{-1/3}$ in equation ((ref)) yields \[ \left( Nh_{N}^{k_{1}+k_{2}}\right) ^{2/3}\mathbb{E}\left[ \phi_{Ni}\left( \beta+\rho\left( Nh_{N}^{k_{1}+k_{2}}\right) ^{-1/3}\right) \right] =\frac{1}{2}\rho^{\prime}\mathbb{V}\rho+o\left( 1\right) \rightarrow\frac {1}{2}\rho^{\prime}\mathbb{V}\rho. \] Similarly, setting $r=\gamma+\delta( N\sigma_{N}^{2k_{1}} ) ^{-1/3}$ in equation ((ref)) gives \[ \left( N\sigma_{N}^{2k_{1}}\right) ^{2/3}\mathbb{E}\left[ \varphi _{Ni}\left( \gamma+\delta\left( N\sigma_{N}^{2k_{1}}\right) ^{-1/3}\right) \right] =\frac{1}{2}\delta^{\prime}\mathbb{W\delta}+o\left( 1\right) \rightarrow\frac{1}{2}\delta^{\prime}\mathbb{W\delta}. \] $\mathbb{H}_{1}$ and $\mathbb{H}_{2}$ can be obtained in the same way as in KimPollard1990. We omit some similar yet tedious details and refer interested readers to Section 6.4 in KimPollard1990. To get $\mathbb{H}_{1},$ we let $\upsilon_{N}\equiv( Nh_{N}^{k_{1}+k_{2}} ) ^{1/3}$ and define \begin{align*} \mathbb{L}\left( \rho_{1}-\rho_{2}\right) & \equiv\lim_{N\rightarrow \infty}\upsilon_{N}\mathbb{E}\left\{ h_{N}^{k_{1}+k_{2}}\left[ \phi _{Ni}\left( \beta+\rho_{1}\upsilon_{N}^{-1}\right) -\phi_{Ni}\left( \beta+\rho_{2}\upsilon_{N}^{-1}\right) \right] ^{2}\right\} ,\\ \mathbb{L}\left( \rho_{1}\right) & \equiv\lim_{N\rightarrow\infty} \upsilon_{N}\mathbb{E}\left\{ h_{N}^{k_{1}+k_{2}}\left[ \phi_{Ni}\left( \beta+\rho_{1}\upsilon_{N}^{-1}\right) -\phi_{Ni}\left( \beta\right) \right] ^{2}\right\} ,\text{ and}\\ \mathbb{L}\left( \rho_{2}\right) & \equiv\lim_{N\rightarrow\infty} \upsilon_{N}\mathbb{E}\left\{ h_{N}^{k_{1}+k_{2}}\left[ \phi_{Ni}\left( \beta+\rho_{2}\upsilon_{N}^{-1}\right) -\phi_{Ni}\left( \beta\right) \right] ^{2}\right\} . \end{align*} Since $\phi_{Ni}\left( \beta\right) =0,$ \begin{equation} \mathbb{H}_{1}\left( \rho_{1},\rho_{2}\right) =\frac{1}{2}\left[ \mathbb{L}\left( \rho_{1}\right) +\mathbb{L}\left( \rho_{2}\right) -\mathbb{L}\left( \rho_{1}-\rho_{2}\right) \right] . \end{equation} We calculate $\mathbb{L}$ as follows. Notice that \begin{align*} & \mathbb{E}\left[ h_{N}^{k_{1}+k_{2}}\mathcal{K}_{h_{N}}(X_{ijts} ,W_{its})^{2}\right] \\ = & \int\frac{1}{h_{N}^{k_{1}+k_{2}}}\left[ \Pi_{\iota=1}^{k_{1}}K\left( \frac{X_{ijts,\iota}}{h_{N}}\right) \Pi_{\iota=1}^{k_{2}}K\left( \frac{W_{its,\iota}}{h_{N}}\right) \right] ^{2}\text{d}F_{X_{ijts},W_{its} }=O(1),\\ & \mathbb{E}\left[ h_{N}^{k_{1}+k_{2}}\mathcal{K}_{h_{N}}(X_{ijts} ,W_{its})\mathcal{K}_{h_{N}}(X_{ij^{\prime}t^{\prime}s^{\prime}} ,W_{it^{\prime}s^{\prime}})\right] \\ = & \int\int\frac{1}{h_{N}^{k_{1}+k_{2}}}\Pi_{\iota=1}^{k_{1}}K\left( \frac{X_{ijts,\iota}}{h_{N}}\right) \Pi_{\iota=1}^{k_{2}}K\left( \frac{W_{its,\iota}}{h_{N}}\right) \\ & \cdot\Pi_{\iota=1}^{k_{1}}K\left( \frac{X_{ij^{\prime}t^{\prime}s^{\prime },\iota}}{h_{N}}\right) \Pi_{\iota=1}^{k_{2}}K\left( \frac{W_{it^{\prime }s^{\prime},\iota}}{h_{N}}\right) \text{d}F_{X_{ijts},W_{its}}\text{d} F_{X_{ij^{\prime}t^{\prime}s^{\prime}},W_{it^{\prime}s^{\prime}}}\\ = & o\left( 1\right) ,\text{ for any }\left( j,t,s\right) \neq\left( j^{\prime},t^{\prime},s^{\prime}\right) , \end{align*} and \begin{align*} & \phi_{Ni}\left( \beta+\rho_{1}\upsilon_{N}^{-1}\right) -\phi_{Ni}\left( \beta+\rho_{2}\upsilon_{N}^{-1}\right) \\ = & \sum_{t>s}\sum_{d\in\mathcal{D}}\left\{ \mathcal{K}_{h_{N}} (X_{i2ts},W_{its})Y_{idst}\left( -1\right) ^{d_{1}}\left( 1\left[ X_{i1ts}^{\prime}\left( \beta+\rho_{1}\upsilon_{N}^{-1}\right) >0\right] -1\left[ X_{i1ts}^{\prime}\left( \beta+\rho_{2}\upsilon_{N}^{-1}\right) >0\right] \right) \right. \\ & \left. +\mathcal{K}_{h_{N}}(X_{i1ts},W_{its})Y_{idst}\left( -1\right) ^{d_{2}}\left( 1\left[ X_{i2st}^{\prime}\left( \beta+\rho_{1}\upsilon _{N}^{-1}\right) >0\right] -1\left[ X_{i2st}^{\prime}\left( \beta+\rho _{2}\upsilon_{N}^{-1}\right) >0\right] \right) \right\} . \end{align*} Thus, the interaction terms after expanding the square in $\mathbb{L}$'s are asymptotically negligible, and we only need to focus on the squares of each term above. We now derive the square of the first term multiplied by $h_{N}^{k_{1}+k_{2}}$ and the rest follow similarly. \begin{align*} & \mathbb{E}\left\{ h_{N}^{k_{1}+k_{2}}\left[ \mathcal{K}_{h_{N}} (X_{i2ts},W_{its})Y_{idst}\left( -1\right) ^{d_{1}}\left( 1\left[ X_{i1ts}^{\prime}\left( \beta+\rho_{1}\upsilon_{N}^{-1}\right) >0\right] -1\left[ X_{i1ts}^{\prime}\left( \beta+\rho_{2}\upsilon_{N}^{-1}\right) >0\right] \right) \right] ^{2}\right\} \\ = & \mathbb{E}\left\{ \mathbb{E}\left[ \left. \left\vert Y_{idst} \right\vert \left\vert 1\left[ X_{i1ts}^{\prime}\left( \beta+\rho _{1}\upsilon_{N}^{-1}\right) >0\right] -1\left[ X_{i1ts}^{\prime}\left( \beta+\rho_{2}\upsilon_{N}^{-1}\right) >0\right] \right\vert \text\right\vert X_{i2ts},W_{its}\right] h_{N}^{k_{1}+k_{2}}\mathcal{K}_{h_{N} }^{2}(X_{i2ts},W_{its})\right\} \\ = & \mathbb{E}\left[ \left. \left\vert Y_{idst}\right\vert \left\vert 1\left[ X_{i1ts}^{\prime}\left( \beta+\rho_{1}\upsilon_{N}^{-1}\right) >0\right] -1\left[ X_{i1ts}^{\prime}\left( \beta+\rho_{2}\upsilon_{N} ^{-1}\right) >0\right] \right\vert \text\right\vert X_{i2ts} =0,W_{its}=0\right] \\ & \cdot f_{X_{2ts},W_{ts}}\left( 0,0\right) \left[ \int K^{2}\left( u\right) \text{d}u\right] ^{k_{1}+k_{2}}+O\left( h_{N}\right) \\ = & \mathbb{E}\left\{ \mathbb{E}\left[ \left\vert Y_{idst}\right\vert \text|X_{i1ts},X_{i2ts}=0,W_{its}=0\right] \right. \text\\ & \cdot\left\vert 1\left[ X_{i1ts}^{\prime}\left( \beta+\rho_{1} \upsilon_{N}^{-1}\right) >0\right] -1\left[ X_{i1ts}^{\prime}\left( \beta+\rho_{2}\upsilon_{N}^{-1}\right) >0\right] \right\vert \text|\left. X_{i2ts}=0,W_{its}=0\right\} \\ & \cdot f_{X_{2ts},W_{ts}}\left( 0,0\right) \left[ \int K^{2}\left( u\right) \text{d}u\right] ^{k_{1}+k_{2}}+O\left( h_{N}\right) \\ = & \int\kappa_{dts}^{\left( 4\right) }\left( x\right) \left\vert 1\left[ x^{\prime}\left( \beta+\rho_{1}\upsilon_{N}^{-1}\right) >0\right] -1\left[ x^{\prime}\left( \beta+\rho_{2}\upsilon_{N}^{-1}\right) >0\right] \right\vert f_{X_{1ts}|\left\{ X_{2ts}=0,W_{its}=0\right\} }\left( x\right) \text{d}x\\ & \cdot f_{X_{2ts},W_{ts}}\left( 0,0\right) \bar{K}_{2}^{k_{1}+k_{2} }+O\left( h_{N}\right) , \end{align*} where $\kappa_{dts}^{\left( 4\right) }\left( x\right) \equiv \mathbb{E}\left[ \left\vert Y_{idst}\right\vert \text{ }|X_{i1ts} =x,X_{i2ts}=0,W_{its}=0\right] $ and $\bar{K}_{2}\equiv\int K^{2}\left( u\right) \text{d}u$. By Assumption P9, $\upsilon_{N}h_{N}= ( Nh_{N} ^{k_{1}+k_{2}} ) ^{1/3}h_{N}\rightarrow0,$ so the bias term is asymptotic negligible. To calculate the above integral, we follow KimPollard1990 and decompose $X_{1ts}=a\beta+\bar{X}_{1ts},$ with $\bar{X}_{1ts}$ orthogonal to $\beta.$ We use $f_{X_{1ts}}\left( a,\bar{x}\right) $ to denote the density of $X_{1ts}$ at $X_{1ts}=a\beta+\bar{x}$. Using the results in KimPollard1990, \begin{align*} & \lim_{N\rightarrow\infty}\upsilon_{N}\cdot\int\kappa_{dts}^{\left( 4\right) }\left( x\right) \left\vert 1\left[ x^{\prime}\left( \beta +\rho_{1}\upsilon_{N}^{-1}\right) >0\right] -1\left[ x^{\prime}\left( \beta+\rho_{2}\upsilon_{N}^{-1}\right) >0\right] \right\vert f_{X_{1ts} |\left\{ X_{2ts}=0,W_{ts}=0\right\} }\left( x\right) \text{d}x\\ = & \int\kappa_{dts}^{\left( 4\right) }\left( \bar{x}\right) \left\vert \bar{x}^{\prime}\left( \rho_{1}-\rho_{2}\right) \right\vert f_{X_{1ts} |\left\{ X_{2ts}=0,W_{ts}=0\right\} }\left( 0,\bar{x}\right) \text{d} \bar{x}. \end{align*} To summarize, we have \begin{align*} & \lim_{N\rightarrow\infty}\upsilon_{N}\mathbb{E}\left\{ h_{N}^{k_{1}+k_{2} }\left[ \mathcal{K}_{h_{N}}(X_{i2ts},W_{its})Y_{idst}\left( -1\right) ^{d_{1}}\left( 1\left[ X_{i1ts}^{\prime}\left( \beta+\rho_{1}\upsilon _{N}^{-1}\right) >0\right] -1\left[ X_{i1ts}^{\prime}\left( \beta+\rho _{2}\upsilon_{N}^{-1}\right) >0\right] \right) \right] ^{2}\right\} \\ = & \int\kappa_{dts}^{\left( 4\right) }\left( \bar{x}\right) \left\vert \bar{x}^{\prime}\left( \rho_{1}-\rho_{2}\right) \right\vert f_{X_{1ts} |\left\{ X_{2ts}=0,W_{ts}=0\right\} }\left( 0,\bar{x}\right) \text{d} \bar{x}f_{X_{2ts},W_{ts}}\left( 0,0\right) \bar{K}_{2}^{k_{1}+k_{2}}. \end{align*} Similarly, we have for the second term \begin{align*} & \lim_{N\rightarrow\infty}\upsilon_{N}\mathbb{E}\left\{ h_{N}^{k_{1}+k_{2} }\left[ \mathcal{K}_{h_{N}}(X_{i1ts},W_{its})Y_{idst}\left( -1\right) ^{d_{2}}\left( 1\left[ X_{i2st}^{\prime}\left( \beta+\rho_{1}\upsilon _{N}^{-1}\right) >0\right] -1\left[ X_{i2st}^{\prime}\left( \beta+\rho _{2}\upsilon_{N}^{-1}\right) >0\right] \right) \right] ^{2}\right\} \\ = & \int\kappa_{dts}^{\left( 5\right) }\left( \bar{x}\right) \left\vert \bar{x}^{\prime}\left( \rho_{1}-\rho_{2}\right) \right\vert f_{X_{2ts} |\left\{ X_{1ts}=0,W_{ts}=0\right\} }\left( 0,\bar{x}\right) \text{d} \bar{x}f_{X_{1ts},W_{ts}}\left( 0,0\right) \bar{K}_{2}^{k_{1}+k_{2}}, \end{align*} where \[ \kappa_{dts}^{\left( 5\right) }\left( x\right) \equiv\mathbb{E}\left[ \left\vert Y_{idst}\right\vert \text{ }|X_{i2ts}=x,X_{i1ts}=0,W_{its} =0\right] . \] Therefore, \begin{align*} \mathbb{L}\left( \rho_{1}-\rho_{2}\right) & =\lim_{N\rightarrow\infty }\upsilon_{N}\mathbb{E}\left\{ h_{N}^{k_{1}+k_{2}}\left[ \phi_{Ni}\left( \beta+\rho_{1}\upsilon_{N}^{-1}\right) -\phi_{Ni}\left( \beta+\rho _{2}\upsilon_{N}^{-1}\right) \right] ^{2}\right\} \\ & =\sum_{t>s}\sum_{d\in\mathcal{D}}\left\{ \int\kappa_{dts}^{\left( 4\right) }\left( \bar{x}\right) \left\vert \bar{x}^{\prime}\left( \rho _{1}-\rho_{2}\right) \right\vert f_{X_{1ts}|\left\{ X_{2ts}=0,W_{ts} =0\right\} }\left( 0,\bar{x}\right) \text{d}\bar{x}f_{X_{2ts},W_{ts} }\left( 0,0\right) \right. \\ & \left. +\int\kappa_{dts}^{\left( 5\right) }\left( \bar{x}\right) \left\vert \bar{x}^{\prime}\left( \rho_{1}-\rho_{2}\right) \right\vert f_{X_{2ts}|\left\{ X_{1ts}=0,W_{ts}=0\right\} }\left( 0,\bar{x}\right) \text{d}\bar{x}f_{X_{1ts},W_{ts}}\left( 0,0\right) \right\} \bar{K} _{2}^{k_{1}+k_{2}}. \end{align*} Finally, by equation ((ref)), \begin{align*} & \mathbb{H}_{1}\left( \rho_{1},\rho_{2}\right) \\ = & \frac{1}{2}\sum_{t>s}\sum_{d\in\mathcal{D}}\left\{ \int\kappa _{dts}^{\left( 4\right) }\left( \bar{x}\right) \left[ \left\vert \bar {x}^{\prime}\rho_{1}\right\vert +\left\vert \bar{x}^{\prime}\rho _{2}\right\vert -\left\vert \bar{x}^{\prime}\left( \rho_{1}-\rho_{2}\right) \right\vert \right] f_{X_{1ts}|\left\{ X_{2ts}=0,W_{ts}=0\right\} }\left( 0,\bar{x}\right) \text{d}\bar{x}f_{X_{2ts},W_{ts}}\left( 0,0\right) \right. \\ & +\left. \int\kappa_{dts}^{\left( 5\right) }\left( \bar{x}\right) \left[ \left\vert \bar{x}^{\prime}\rho_{1}\right\vert +\left\vert \bar {x}^{\prime}\rho_{2}\right\vert -\left\vert \bar{x}^{\prime}\left( \rho _{1}-\rho_{2}\right) \right\vert \right] f_{X_{2ts}|\left\{ X_{1ts} =0,W_{ts}=0\right\} }\left( 0,\bar{x}\right) \text{d}\bar{x}f_{X_{1ts} ,W_{ts}}\left( 0,0\right) \right\} \bar{K}_{2}^{k_{1}+k_{2}}. \end{align*} The same arguments can be applied to obtain $\mathbb{H}_{2}$ as \begin{align*} \mathbb{H}_{2}\left( \delta_{1},\delta_{2}\right) = & \frac{1}{2}\sum _{t>s}\int\kappa_{\left( 1,1\right) ts}^{\left( 6\right) }\left( \bar {w}\right) \left[ \left\vert \bar{w}^{\prime}\delta_{1}\right\vert +\left\vert \bar{w}^{\prime}\delta_{2}\right\vert -\left\vert \bar{w}^{\prime }\left( \delta_{1}-\delta_{2}\right) \right\vert \right] f_{W_{ts}|\left\{ X_{1ts}=0,X_{2ts}=0\right\} }\left( 0,\bar{w}\right) \text{d}\bar{w}\\ & \cdot f_{X_{1ts},X_{2ts}}\left( 0,0\right) \bar{K}_{2}^{2k_{1}}, \end{align*} where $W_{ts}=a\gamma+\bar{W}_{ts}$ with $\bar{W}_{ts}$ orthogonal to $\gamma $, $f_{W_{ts}}\left( a,\bar{w}\right) $ denotes the density of $W_{ts}$ at $W_{ts}=a\gamma+\bar{W}_{ts},$ and \[ \kappa_{\left( 1,1\right) ts}^{\left( 6\right) }\left( w\right) \equiv\mathbb{E}\left[ \left\vert Y_{i\left( 1,1\right) ts}\right\vert \text{ }|W_{its}=w,X_{i1ts}=0,X_{i2ts}=0\right] \text{.} \] \end{proof}